i 



TC 





!^ 





Class_ JG I (^ o 
Book 'V 7^ 
Cojjyiigtit N?^ 



COPyRIGHT DEPOSm 



NOTES ON HYDRAULICS 

PREPARED FOR THE USE OF STUDENTS 

IN THE SHORTER COURSES 

AT THE 

MASS. INSTITUTE OP TECHNOLOGY 

BY 

GEORGE E. RUSSELL 

1907 



- Copyright, 1907, - 
by G.E.Russell 



,fcO 



^^l^^ 



LIBRARY of C0N6KE3S, 
Two Copies BecBivefl 

DEC 2 iy07 

Oopyriffii tnjry 

/Y*t/ 3* '?*? 

claSS A- "'^<^' *<■ 

''cOPt B. 






*^ 






Prefatory Note 



^ In view of the fact that this set of notes may fall into 

^ hands other than those for which it was intended, the author wish- 

^ es to state that it is not to be considered in any sense a com- 

T> plete treatise of the subject, or even of that part of the subject 

"^ touched upon. The aim has been to prepare a text, suitable for 

^ students taking shorter courses in Hydraulics, which will permit 

^5^ of revision and expansion after a year's use. The absence .of 
many familiar and importsmt discussions is thus explained. 

Mass. Inst. Tech. 

Boston, 1907. 



^.? 



NOTES ON HYDRAULICS 

CHAPTER I 
-HYDROSTATIOS - 

i_. Definitiona. - As the study of the laws governing the 
conditions of fluid bodies is an application of the laws of Stat- 
ics and Dynamics, such a study may be rightly termed the Mechanics 
of Fluids. If we confine ourselves to the class of fluids known 
as liquids, and further limit our discussion to the one liquid. 
Water, we shall have as two natural divisions of the subject,- Hy- 
dro-Statics , a discussion of water at rest, and H ydro-Dynamics , a 
study of water in motion. The general study of these two divis- 
ions is commonly referred to. as Hydraulics, although the word or- 
iginally signified the flow of water in a pips. 

While the use of those terms limits us to the one liquid. 
Water, it is quite evident that in the case of liquids whose physi- 
cal properties are similar to those of water, the same laws and rea- 
sonin gs can be made to apply. 

2. Water .- Water is not a perfect fluid although for en- 
gineering purposes it is quite permissable to assiome it so. By a 
perfect fluid we mean a fluid between whose particles there is no 
friction. In such a fluid the pressure of one particle upon an- 
other must be wholly normal and therefore the pressure upon any 
plane or surface in the water must also be normal. 

Water is often said to be non-compressible but this is a 
mistake. However, its decrease in voliome under all ordinary pres- 
sures is so small that we may neglect it in all future discussion. 
The experiments of Grassi indicate that a pressure of one atmosphere 
(14.7 lbs. per sq. in.) will cause a diminution in volume expressed 
by the cubic-coefficient 0.00005. 

The weight of water, or its heaviness per unit of volume, 
varies with its temperature and its purity. As one would expect, 
water near the boiling point is much lighter than at 39.3° Pahr. 
where it has its maximum density. Salt water, too, is known to be 
heavier than fresh, although the difference is not great. The 
heaviness of a cubic foot being denoted by w, the following values 
will be found sufficient for all ordinary calculations .- 

Fresh water w = 62.5 lbs. 
Salt water 64.0 " 

Mercury 848.7 " 

Mercury is mentioned as its use in measuring water pres- 
sure is very convenient. 



-2- 



5. Units of Measure. - The elements of distemce, force, 
and time enter largely Into our discussion and will be measured by 
the units of the foot-po\ind-second system. The foot is the com- 
mon English foot of 12 inches and the pound is the avoirdupois 
pound of 16 ounces. These are the units employed in all Hydrau- 
lic foiTnulae found in these notes and it is essential to see that 
all data is reduced to these units before maJcing substitution in 
the formulae. In this connection the following relations will be 
found useful. - 

1 cu. ft. = 7.48 U.S. galls. 

1 U.S. gall. = 231 cu. inches 

1 U.S. gall. = 0.8331 Imperial galls. 

The U.S. Gallon is in common use in this country while the Imperi- 
al Gallon is used in Great Britian. 

As the motion of water in moat cases is due to the action 
of gravity,, the quantity g will often appear in our work. By g 
we represent the acoeleraTion, due to gravity, of a freely falling 
body. It is the change in velocity during one second of time and 
consequently will be expressed in feet per second per second. 
Its value of course changes from point to point on the earth's sur- 
face, but a mean value may be stated to be 32.2 ft. per sec. per 
sec^ 

4. Intensity of Pressure. - By intensity of pressure we 
denote the pressure per unit of area. Water pressure in this 
country is commonly measured in pounds per square inch. If by dA 
we represent a very small part of an area A, and by dP the total 
pressure upon it, then the intensity of pressure is 

p = _dP_ 
dA 

If the intensity be the same for the entire area A, we may write 
the total pressure on A as 



sfp.dA = 



PA 



from which 



p = _Z- 

A (1) 

The student will see that if p is not the same over the entire a- 
rea, formula (1) gives an avera ge value of p. 

Atmosphere, being a perfect fluidj exerts a normal pres- 
sure upon all objects with which it comes in contact. The value 
of its intensity, Pg^, at sea level is found by experiment to be 



-3- 



about 14.7 per sq. inch, 



(Barometer at 30") 



5. Pressure on Submerged Surfaces . - { 1 ) A Horizontal 
Plane . (Fig"^ 1) Let A-B represent the edge of the plane lying a 
distance h below the water surface. The water surface is in a 

free state being pressed upon by at- 
mosphere. Consider the plane and its 
superimposed prism of water A3CD to be 
removed from the rest of the liquid 
(Pig. 2) and arrows to be substituted 
for the forces that previously pressed 



t 



upon them. The whole is in 

equilibriiiin under the forces 

acting, and from SV = we 
have 



pA = PaA + W 
But 

W = wAh 
Therefore 

pA = Pa,A + wAh or p = Pa + wh 






— 




— 




a 


J, 




/J 


«.. 






1. 






(2) 



The other forces acting have no vertical components, so 
do not enter into the discussion. 

(2) Any Oblique Plane. - Since water transmits pressure 
equally in all directions and the intensity of pressure on a hori- 
zontal plane passing through a point distant h from the surface is 
P .( ~ Pa ■*■ "f^) 5 ^^ follows that the intensity at this point on any 
plane passing through the point will also be 

P = Pa + w^- 

Note .- Inasmuch as p varies only and directly with 
the depth h, it follows that the—value of p is constajit over any 
horizontal plane. ~ 



6. Relation between Pressures at Different Depths. - In 
Fig. (3) the reservoir M is connected by the tube C-D with the res- 



_4- 



ervoir at the lower level N. All portions of the tube are filled 
with water as they lie below tiie level at M-0 . 




f/-^.^. 



The weight W on the piston at N is just heavy enough to maintain 
the water in M at a constant level. 

For the pressure at any point such as A we may write 

p-|^ = wh-]^ (Neglecting atmosphere) 

and for a point B a,t a depth hg below A the pressure is evidently 
P]^ increased by that due to a column hg in height. Therefore 



P2 = Pi 



+ whc 



This value of pg -must hold good for points C and D as they lie in 
the same horiz'ontal plane. For a point E we have 

P3 = pg + wh3 

for, while the points D and E are not in the same vertical, we may, 
if we please, follow the increase in pressure downward by dropping 
in ve'rtical lines from one horizontal plane to another (as shown 
by the broken line D-E) and arrive at the same result. 
As for the pressure at N, beneath the piston. 



P4 = P3 



- wh/ 



which may be written by inspection of the three preceding equations 

P4 = wh]^ + whg + whg - wh^ 
= w (h^ + hg + h3 - h^ 
or 



-5- 



7 . Atmospheric Pressure . - Absolute Pressure . - Pressure- 
Head .- In the preceding problem we have neglected atmospheric 
pressure. As the water transmits this pressure from the water 
surface equally in all directions and Without loss, we may if we 
please, add pg. to the value of p^ as last obtained. However, In 
this particular problem, p^^ is the pressure on each unit area of 
the tinder side of the piston at N which is open to atmosphere on 
its upper side. Therefore as far as the effectiveness of the 
water pressure in sustaining the weight W is concerned, pg^ need 
not enter into the discussion as it acts on both sides of the pis- 
ton and produces no resultemt force. With this elimination the 
value of P4 will measure the excess of pressure over or " above the 
atmosphere." We call this relative pressure . With p^ included 
in Pa the pressure is measured "above the vacuum." This we des- 
ignate as absolute p ressure , 

Although in much of our work absolute pressure is not 
used, there will arise cases where it must be employed and a state- 
ment that it is used will preclude the possibility of serious er- 
ror. In measuring pressures use is often made of the Bourdon gage 
which registers relative pressures. 

In the solution of practical problems it is important to 
see whether or not the effect of atmosphere must be considered, 
and the student is warned against its elimination without careful 
inspection of the problem at hand. 

In deriving the expression for p we used the distance h 
which was measured vertically downward from the water surface. 
It is quite common to designate this distajice as the Head on the 
point in question. From p = wh we may write 

w - h 

As h is the head that gave the pressure p it is known as the "Pres- 
sure-Head." The same name is given to Tts equal, P 

w ' 
If the pressure p be atmospheric, the pressure-head will 
be 

1 4.7 X 14 4 ,^ ^^ 

— 6275 — = 34 ft. nearly. 

That is, atmospheric pressure is equal to that produced by a head " 
of water measuring 34 feet. 



8. Total Pressure on a Plane Surface.- (Fig. 4). Here 
the surface is represented by A-B-C, lying in the plane of MON. 
If the plane of A-B-C be produced upward to intersect the water 

surface, it will cut it in the 
line 0-M, and the angle between 
these two surfaces we will call or. 
Selecting a very small part, dA, 
of the total, area, so small that 
the pressure over it is of uniform 
intensity, we have as the total 
pressure upon it 

dP = p.dA = wh.dA 

For total pressure on A-B-C we 
have 




P =[wh.dA =|w.x sinoc.dA = w sineo^jx. 



dA 



The integral of x.dA is the moment of the area A-B-C about 0-M as 
an axis, and may therefore be written x^A where x^ is the distance 
from the axis to the center of gravity of the area. We may there- 
fore write 

P = w sineo( x^ A 
or 



P = A wh. 



(3) 



■where ho is the head on the center of gravity. Expressing this 
result in words,- The total pressure on an immersed area is the 
product of that area, the weight of a cubic unit of water and the 
head u p on its centre of gravit y . 

9. Total Pressure on a Curved Surface. - It can be easily 
shown that the above theorem applies with equal exactness to any 
surface, be it plane, curved or irregular. However, the total 
pressure on surfaces of the last two classes, is of little or no 
practical value to the engineer. It would be the sum of a system 
of forces all acting in different directions, and in general we 
know that such a system cannot be replaced by a single resultant. 

More often is' it desired in the case of curved surfaces 
to find a component of normal pressure in some fixed direction. 
(See Art. 13) 

10. Center of Pressure .- An Immersed plane surface is 
pressed upon by a system of parallel forces. Infinite in number, 
which may be replaced by a single resultant force. The point on 



-7- 



the surface at' which this acta we will call the " Center of Pres - 
sure ." As noted in the previous paragraph, the pressures on a 
curved surface do not form a parallel system, hence they cannot in 
g eneral , be reduced to a single force. 

The case of the plane surface may be represented by Fig. 
4, The resultant of all the pressures on the area A-B-C is sup- 
posed to be acting at the point c.p. and we are to determine first 
its distance from 0-M, the line of intersection of the surface 
plane and the plane of the water surface . 

Considering a very small area dA we have as the total 
pressure upon it, 



dP = dA.wh 



(a) 



and its moment about 0-M is 

dP.x = dA.wh. X (b) 

If in this way the moment of the pressure on each elementary area 
be found, we may place their si^ra equal to the moment of the re- 
sultant pressure by its arm x^ . That is, 



P Xc = 



=JdP.: 



or, from (a) and (b) 



Prom Pig. 4, 



.qI dA.wh = JdA.wh, 



h = x.sine o< 



X (c) 



which substituted in (c) gives 

w.sinecY.x^ Ix.dA = w.sineorjx?dA 



or 



Xo = 



Jx?dA_ _ i 



'c "/x.dA ~ S (4) 

The integral of x?dA will be recognized as the Moment of 
Inertia of the area, while the integral of x.dA is the expressed 
moment of the area about the chosen axis, or its Statical Moment , 

It is evident that Xq measures only the distance of the 
center of pressure down from 0-M and does not fix it in a lateral 
position. The latter may be found by taking moments of the pres- 
sures about another axis, in the same plane as A-B-C but at right 
angles to 0-M, thus obtaining another co-ordinate, y^, for the 



-8- 



, point. Some special oases permit of a more easy solution as will 
be noted in the following paragraphs . 

In general the engineer has only to do with the vertical 
position of the center of pressure. 



Horizontal Surfaces .- If the surface be horizontal, the 
the head on all points being the same, the forces are all equal, 
and the center of pressure will lie at the center of gravity. 

11. Examples. - The surfaces most commonly met with in 
engineering are the rectangle, triangle and circle. For these we 
will now derive general equations for locating the c. of p. Ref- 
erence to formula (4) shows that the position of the c. of p. is 
independent of the angle ex. (Pig. 4) provided it has any value oth- 
er than zero. Hence we may rotate the area A-B-C about 0-M with- 
out the c. of p. chajiging. If c>< = 90 , the h for each elementary 
area becomes its x also, and it will be a matter of convenience if 
we assiime our surfaces vertical. 

Rectan gle.- Here we will take our elementary area as a 
horizontal strip. Then all parts of it will have the same x. 

_/x?dA 



X-, =• 




/x.dA 



where dA = b.dx 



-\h x?dx 






,b X . dx 



hj - hi 
- hi 



he 



If hi = 0, or the upper edge of the 
rectangle be in the water surface. 

This is very useful and should be re- 




y^6-. 



Triangle .- Base 
Horizontal . Vertex Up 

dA = u.dx i 

From similar triangles 

u _ x-h] 



h2 



% 



_ b(x-hi) 

or u = — i ii- 

h2-hi 



-9- 





^=- n 






'z 






U 


b-A 







/v^.G. 



Tx^dA x2{x-hi) _b_ 



/ 



x.dA 



h2-hi 



dx 



b 
_ h2-hi 



_b_ 



r^ , . s b 



dx 



112 



hi 4 



x^_ hix^ 



n2 

hi 3 ' "2~ 



or 



X = 1 (hp-hi )^(5h2+2.hih2+hi) _ i 5h2+ghih2+hi 
*" ' 2 (^g_h,)2(2h2•^hl) 2- 2h2+hi 



If h^ = or the vertex lie in the water surface then 

_ 3 



^c = t-^ 



Trian g le.- Base Horizontal . Vertex Down. - By the same 
method as before we may arrive at the .result 



/7y./ 



Xo = 



= 1 



3hj^ + ghghi 




2hi + hg 



If h;^.= 0, Xc = I d 



Note.- In the case of the rectangle 
and triangles our elementary areas 
had their length parallel to the sur- 
face. This means that the pressure 
on each strip was uniformly distrib- 
uted and might be replaced by a sin- 
gle force acting at the middle of the 
strip. 
Conceiving the whole area to be subdivided into elements, each 
having its resultant force, we find these latter lying on a medial 
line, as does also their resul'tant. In these cases the c. of p. 
is exactly located by formula (4) and the medial line. 

Corollar y.- If an area have an axis of symmetry at right 



-10- 



angles to the moment axis, the c. of p. will lie upon it. 

Circle. - Here let h be the head on the center of the 



circle . 




^c = 
or 

Xo = 



_ Jx^dA _ Iq+A x< 



/x.dA 

A. 



+ TTr^h 



A X, 



i2i,2 



ir r^h 



= h + 



4h 



/y^. 6. 



12. Relation between Center of Gravit y and Center of 
Pressure .- That the c. of p. always lies below the c. of g. may be 
easily seen from Fig. 9a. 



C-- 




/y^.9. 




/y^. SI?. 



If A-B represent the edge of any irregular plane area, we may erect 
at A a perpendicular AC which will represent the intensity of pres- 
sure at that point. If we intersect the water surface with the 
plane produced, and draw OC, this line will represent the variance 
of pressure as we go downward. If all the pressures were equal, 
their resultant would act at the c. of g. of the area; but with 
the trapezoidal distribiition of pressures, as shown, the resultant 
must act through the c. of g. of the trapezoid, thus bringing the 
c. of p. below the area's c. of g. 

If the area be a rectangle with its upper edge in the 
water surface (Fig. 9b), the distribution becomes triangular and 
the resultant acts through the c. of g. of the triangle: i.e. two- 
thirds of the distance down, as was previously proved. 



-11- 



15. Pressure In a Given Direction .- Instead of finding 
the total normal pressure on a surface, it may be desired to find 
a component of this pressure parallel to some fixed direction. - 
There arise several distinct cases. 

(1) Plane Surface .- In Fig. 10, P-^^ may represent the re- 
quired component, making ein angle of with the normal pressure P. 

The other component of P is, of 
course, perpendicular to the required 
component . 




For the value of P^we have 
P-j^ = Ecosoc = A wh cosof 



(a) 



^/r" 



f,^./o. 



The area A is represented by C-D in 
edge-view, and if Ci-Di be the pro- 
jection of A on a plane perpendicular to P]^, we have as a value of 
this projected area, 



from which and (a) 



At = Acosof 



Pi = A^who 



<5) 



or V-^ is the product of the projected area and the intensity of 



of g. of the original area.- 



Hence the follow- 



pressure at the c, 
ing 

Theorem . - To find the component of pressure parallel to 
a given direction, project the area upon a plane perpendicular to 
the component and multiply the projection by the intensity of pres- 
sure on the c. of g. of the original area. 

(2) Curved and Irre g ular Surfaces .- (Fig. 11) The fol- 
lowing figure and demonstration are 
given for the special case of a sur- 
face of single curvature, although 
the same reasoning and results would 
be obtained from the consideration 
of an irregular or double curved sur- 
face. Again C-D will represent the 
edge -view of our surface . We may 
imagine it to be such a surface as 
might be formed from a bent iron 
sheet. If we consider a small ele- 
mentary area dA, the normal pressure 
upon it is 




..^'—t^' 



/^.// 



dP = dA«wh and dP^ = dP.co3<5c= dA.wh cosoi 



Evidently the total value of V\ may be expressed as 



-12- 



,=;, 



dA«wh cos ex 



(6) 



where h and cos of are both variable, having In general , no relation. 
The expreaaion cannot then be simplified, but a graphical represen- 
tation may be made. If on a plane perpendicular to P-j^ the area 
C-D be projected, we obtain an area representing the integral of 
dA'COsi^. On each projection of a dA we may imagine a normal or- 
dinate to be erected equal to the h for the corresponding dA in 
the real surface. Such an ordinate is x-y. We now obtain a vol- 
ume representing JdA.h.cosot and if we consider it to be of water, 
its weight will be the value of Pi. 

(3) Special Cases .- (a) If in formula (6) the value of h 
be assumed constant we have 



P^ = whfdA.coso<= A^whQ 



which is the same as formula (5) for plane surfaces. Often in 
practical work the head is so large that the atfove assumption can 
be made . 

(b) Again if the conditions are such that the angle o< is 
constant, we obtain 

P^ = w coso^jdA.h = w cosotAho 

P-|^ = A^whQ as before. 



or 



As an illustration of this latter case let the student figure the 

upward vertical component of normal pres- 



^ sure on the submerged cone in Fig. 12, 




/9^./£. 



(c) Perhaps most important of all to the engineer is the 

component of pressure in a Horizon - 
tal direction against an y kind of 
surface . Referring to Pig. 13 it 
will be seen that for this particular 
case, each normal ordinate x-y will 
increase in direct proportion to the 
depth of the ordinate below the sur- 
face. This being so the mean ordi- 
nate hju will be found at the center 
of gravity of the projected area, 
and formula (6) may be written 




Pi = 



A^whrn 



-13- 



or 

The component of pressure in a horizontal direction is 
the same as that upon its vertical projection. 

These general and special cases should be thoroughly 
mastered by the student to avoid serious error in the solution of 
practical problems. 

14. Pipes and Cylindrical Shells under Pressure .- Fig. 
14. This is a practical illustration of the foregoing principles. 

It is desired to find the amount of ten- 
sile stress in a pipe containing water 
under pressure. It will be assumed that 
in any crose-aeotion A-B the intensity of 
pressure is the same at all points of the 
circumference. This is equivalent to as- 
suming the diameter small in comparison 
to the total head on the pipe. In Fig. 
14b, one-half of the pipe has been removed 
Eind the two tensile forces T substituted 
to indicate the action of that half on 
the remaining portion. Any vertical line 
m-n passes through two elementary areas, 
dA, on each of which the pressure is p.dA. 
If we resolve these two pressures into 
horizontal and vertical components we ob- 
tain two equal and opposite vertical for- 
ces, and two equal horizontal forces act- 
ing in the same direction. Evidently 
the sum of the H components acting on all 
the elementary areas of the semi-circum- 
/-■ ij, ference is equal to 2T. (Since for equi- 

^-^- Z-^- librium 2H = 0). 

From Art. 13, the total H component is equal to the pros- 
sure on the vertical projection of our curved area, under the same 
head. If the length of pipe be -unity, the area is d and H = pd. 
Finally, - 




or 



2T = H = pd 

T = £d 
2 



This stress T is distributed over an area t, so that the stress 
per ^xnxt area is 



and 



^t - 2t 
" 2ft f^, 



(7) 

I 

(8) 
(9) 



-14- 



Ordinarily a pipe figured on the basis of this last formula would. 
be too thin to withstand the strains arising from rough handling, 
unequal bearing and _water-hammer, so that in practice it is cus- 
tomary to add an additional ajnount to the thickness obtained from 
(9). 

15. Pressure on Both Sides of a Plane. - In the cases of 
immersed planes previously considered the total pressure on one 

side equalled that upon the other 
as the head on the c. of g. was 
unchanged. Imagine, however, a 
plane such as A-B (Fig. 15) sepa- 
rating the two water levels M and 
N, and let 0-D be an edge view of 
a certain portion of its area. 
The total pressure on any email 
dA will be, for the left hand 
side, 




s 



dP^ = dA.whi 
and for the right hand side 
dp2 = dA.whg 



/?y. /S. 

Their resultant (acting to the right) will be 

dR = dP]^ - dPg = dA.w(hi-h2) = dA.wh 

That la, the resultant pressure on any elementary area is depen- 
dent only on h, the difference in water levels. Each dA of C-D 
then has the same r&sultant pressure and a single resultant ap- 
plied at the c. of g. of C-D may replace these equal pressures. 

This is graphically shown in Pig. 16. The variation of 

pressure over A-B may be represented, 
on the left side, by the straight 
line 0-E, and on the right by A-P. 
The triangles 0-B-E and A-B-P are 
similar and A-B-P and C-D-E are equal 
The trapezoid A-C-E-B represents the 
total pressure on the left of A-B, 
from which we must subtract the tri- 
angle A-B-P or its equal C-D-E in 
order to obtain the resultant pres- 
sure. This will give the rectangle 
of pressure A-B-C-D, which shows the 
resulting pressure to be uniformly distributed over A-B. 



/y^./6. 



3 



F 



-15- 



16. Pressure on Immersed bodies, - Theorem of Archimedes « 
About the year 250 B.C. Archimedes discovered the important law 
that immersed bodies lose a portion of their weight equal to that 

of the volume of water displaced 




^_JU 



by them. The proof is simple. 
AB is any such body through 
which we will pass a vertical 
plane C-D. (Perpendicular to 
plane of paper.) The horizon - 
tal components of total pressure. 
Hi and H2> on the irregular areas 
mAn and mBn must be equal as both 
are measured by the same pro- 
jected area m-n. (Art. 13) If 
the vertical plane C-D were 
passed through the body parallel 
to the plajie of the paper we 
should obtain H3 = H4, these latter forces being at right angles 
to H]^ and Hg . Evidently 2H = for all the normal pressures. 
To investigate the vertical components, assume a vertical prism 
e-f with end areas so small as to give uniform intensity of pres- 
sure upon them. By Art. 13, under these conditions we have 

Vi = dA.wh^ 

dA being the area of the prism's cross-section. 
Similarly we obtain 

V2 = dA.whg 



and the net resulting vertical force on the prism is 

dR = V2 - Vi = dA.w (hg-h^) 

This latter term is the weight of a volume of water equal to that 
of the prism. By a consideration of every elementary prism in 
the body, we may conclude that the body as a whole is acted upon 
by an upward resultant force equal to the weight of the volume of 
water displaced by the body. 

If the body be floating at the surface with only a por- 
tion immersed, the law still holds good, as a consideration of the 
vertical pressures on an elementary prism will show. 

In either case we may conceive of two forces acting on 
the body,- the weight of the body and the "buoyant effort" of the 
water. The weight acts through the center of gravity of the body 
while the "buoyant effort" acts through the center of gravity of 
the volume of displaced liquid. This may be seen by a return to 
an elementary prism. The small upward resultant force acting 



-16- 



upon it being proportional to its volume , it follows -that the fi- 
nal resultant of these elementary resultants must act through the 
center of gravity of the total volume. Tliis point is called the 
" Center of Buoyanc y ." 

17. Depth of Flotation .- To find the depth to which a 
floating body will sink, it is only necessary to remember that 

the weight of the body equals the weight 
of displaced water. Prom the geometri- 
cal relations for any regularly shaped 
body we may then figure the distance. 

Illustration .- Py ramid or cone with 
axis vertical and vertex down. 




Let V = volume of cone 
W = weight " " 
^1— (I I) It 

v'= volume immersed 



per cu. unit 



f/^. /8. 



Then X _ £ 



and V' = l^V 
h3 



The weight of the cone is w'V sj^d the Weight of the water dis- 
placed is w d^.v hence 



h3 



w*v = wd^v and d - hJw* 
h3 ^i 



The ratio w' equals £ the specific gravity of the solid, so that 
finally ^ 



= h"fr 



Problem. - Let the student determine d for the cone or 
pyramid with axis vertical and vertex up . 

18. Stability of Immersed and Floatin g Bodies .- If the 
weight of a body exceeds the buoyant effort it will sink. If it 

- be less, it will, if placed beneath 
the surface, rise to the surface and 
assume a position according to the 
above stated principles. If the 
weight were just equal to the buoy- 
ant effort, it would remain beneath 
the surface wherever placed; or, if 
placed upon the surface, would sink 
until just submerged. Reference to 

fv^/g. Pig. 19 will show that for this lat- 





-17- 



case the body will assiome a position so that its center of gravity 
and the center of buoyancy will lie in the same vertical line. 
This is necessary for equilibrium, as otherwise the weight at the 
c. of g. and the upward force at the c. of b. form a couple which 
will rotate the body. If the c. of g. is below the c. of b., the 
couple will right the body and the equilibrium is stable. If the 
c. of g. be above , the condition is clearly that of linstable equi- 
libriiam, and, if the two points coincide, the body is in equilib- 
rium for all positions. 

If the body floats, the conditions of equilibrium are not 
so easily determined. The position of the center of gravity will 
of course remain tmcheinged, but the volume of displaced water will 
alter its shape as the body is rotated, involving the change in 
position of the center of buoyancy. This is well shown in the 
case of a floating ship. (Fig. 20.) Assiiming the c. of g. 




F/^.20 




/x>^/ 



to be at G sind the c. of b. at C, or below G, it would seem at 
first sight as though the ship were in unstable equilibritim. 
However, consider an outside force to be applied causing the ship 
to roll (Fig. 21.) The position of C is changed to some point 0' 
lying on the side of greater immersion. In the figure shown, the 
line of action of the upward force cuts the axis A-B at a point M 
above the c. of g. Evidently for this particular case the equi- 
libritim is stable as the couple tends to right the ship. If M 
had fallen below G, the action of the couple would have tended to 
upset the ship and the equilibrium would have been unstable. The 
point M is called the " metacenter .** Its position relative to the 
center of gravity becomes then a criterion of stability and it is 
important that we have some means for determining its exact loca- 
tion. It can be quite easily proved that the following equation 
will give the distance h^ measured from the center of gravity G 
(Pig. 21) to the metacenter M. 

Let V = volume of displaced water 

I = least moment of inertia of a water-line section , 
through the body, about an axis through its own 
c. of g. 

c = distance between c.of g. ajid c.of b, 
= CG (Pig. 21) 



-18- 



Then, 



iim = -±C 



(10) 



Here c is minus when the c. of g. is above c. of h. 

For the derivation of this formula and a more detailed 
treatment of the subject the student is referred to an excellent 
article by Prof. I. P. Church in his "Mechanics of Engineering.", 
John Wiley & Sons, New York. 



-19- 

HYDRODYNAMIOS 

CHAPTER I 
BERNOULLIS THEOREM.- ORIFICES 



19. Coefficients .- Water in motion presents problems 
more difficult and uncertain than are found under Hydrostatics 
inasmuch as frictional resistances and disturbances are constantly 
interfering with its movement. This is noticed in the lack of 
perfect agreement between results deduced by theory and those ob- 
tained by experiment. Accordingly it is often necessary to apply 
a numerical factor or coefficient to our theoretical results in 
order to make them agree with actual results. The student is 
warned against the error of thinking that, because we depend upon 
experiment for values for these coefficients, our theory is uncer- 
tain or lacks soundness. 

20. Bemoullis Theorem .- In 1738 Daniel Bernoulli demon- 
strated a general theorem in connection with bodies of steadily 
moving water, which is perhaps the most important in the whole sub- 
ject of Hydraulics. By its use alone a great majority of the 
problems arising in hydro-dynamics may be completely solved. 

The assumption is made that we have a perfect fluid and 
that external frictional resistances are eliminated. The water 
is also considered to be in a state of " stead y flow ." By this is 
meant that in any cross-section taken perpendicular to the direc- 
tion of motion, the velocity of water-particles past any point in 
the section is constant . The pressure and densit y are also con- 
stant. These quantities may change, however, from section tc sec- 
tion. It is with this " stead y flow" that the engineer has mostly 
to deal . 

In Fig. 22 is shown a section of pipe containing water 

under steady flow. The quanti- 
ty* Q> passing any section in a 
given time must be constant, else 
there will be an accumulation of 
water. Let v^ and a^ represent 
respectively the velocity and 
cross-sectional area at A, and 
vg and ag be the same quantities 
for the section at B. Then from 
the above, 




f/^.f^. 



a-lVi = agvg = Q 



-20- 



This is often referred to as the " equation of continuit y." 

Consider now the body of water contained between these 

sections, and the forces acting 
on it. The only working forces 
(those which produce motion) are 
the end pressures at the sections, 
and gravity. The pressures on 
the water by the pipe are normal 
to the direction of motion and, 
not moving, perform no work. 
J Let pj^ be the intensity of pres- 
sure at A and pg that at B. Then 
the three working forces are a^pi, 
agPg and W. If they are allowed to act for a short interval of 
time, the work performed by them will result in an increase in the 
kinetic energy of the water, and an equation may be written be- 
tween the work done and the corresponding increase. Suppose that 
in the time dt A-B nioves to Ar-B ' (Pig. 22.) Then A-Ai = v^dt 
and B-Bj^ = vgdt . 




/y^.^J. 



Work done by ^\p\ = +aipi 
Work done by agPs = -a'2P2 



V]^dt = pj^Qdt 
vgdt = -pgQdt 



This last is negative since it opposes motion. As for the work 
done by gravity during the move from A-B to Aj^ -Bj^ , we may consid- 
er the change in the water's position to have been accomplished by 
moving A-Ai to B-Bi and the work done by gravity would be that of 
making such a change . 

.', Work done by gravity = w.a^^. vi.dt(zi-Zg) = wQdt (z^-zg) 

where zi eind zg represent the heights above any assiomed datum 
plane of the centers of gravity of k-k-y and B-B-j^.. 

We have now to find the change in kinetic energy of A-B. 
Conceive for a moment that all the particles of water between -A 

and B be divided into a great number 
of groups of equal volume, the vol- 
lome of each being that of A-A^ which 
is the first group. Some of these 
are shown in Fig. 24, greatly exag- 
gerated in size. With this assump- 
tion it is easily seen that as group 
a moves forward to displace b, in 
the time dt, its particles gradually 
assume the velocities that those in 
b had at beginning of time dt, and 
thus the group experiences a change in kinetic energy. Simulta- 
neously with this movement each group from a to m advances into the 




f/^.Z4: 




-Sl- 



poeitlon previously held by the immediately preceeding group, and 
m moves to B-Bi at n. The change in energy experienced by any 
group, as £, would be the difference between the origina] kinetic- 
energies of c and d^, inasmuch as c acquires d's velocity and. (hav- 
ing the same mass) its kinetic-energy. The total change in kinet- 
ic energy for the entire body A-B would be obtained therefore by 
adding the dif ferencesbetween contiguous groups. There results 

Total change in K = (kt,-ka^ "*" (^c-kb) + (l^d-I^c) "*" etc . . . . (kj-j-k,,;) 



or 



Change in K = kj^ - k. 



The change is then equal to the difference between the kinetic 
energy of A-A^ emd B-Bj or 

_ w.Q.dt / 2 2. 
|- — (V2 - vi) 

Finally equating work done to chaxige in energy we have 

PlQdt - PgQdt + wQdt (zi-zg) = -2^^^ (v2-vi) (11) 

2g 

2 2 

or vn Pt vp pp 

-i +_+ zi =—+_-+ zp (12) 

2g w -^ 2g w 2 ^J-^; 

This expression constitutes Bemoullis Theorem. It 
states that under steady flow with friction eliminated, the siora of 

2 

g— + ^ + z = a constant quantity (13) 

for all sections. 

Each of these three terms is a linear distance. The 
value P we are already familiar with, it being the head that gives 
rise t5 the pressure p, and known as Pressure-head . The term z 
is simply the height of the particle above any assumed datum and 
will be known as the Potential Head. Lastly v^is Velocity-Head . 
The significajice of the name will appear shortly. The siom of 
these three heads for any particle we will call the Total Head . 
Our last equation may now be thus expressed in words. - 
Tn stead y flovr . without friction , the sijm of velocit y- 
head , pressure-head and potential-head is a constant quantity for 
any p article throughout its course . 

21. Bemoullis Theorem with Friction. - It will be of in- 
terest to note here the effect on Bemoullis Theorem of introduc- 
ing friction. The presence of friction produces resistances or 
rorces which hinder motion. The work of such forces is therefore 



-22- 



negative and Equation (11) must he changed by adding to the left 
hand member a negative quantity to represent this work. The 
quantity must contain a force and a distance, and we may employ 
any force we please provided a proper distance be used. Select- 
ing the force w.Q,dt which already appears in the equation and 
representing the distance by h', we have 

2 2, 



P-j^.Q.dt - Pg.Q.dt + w.Q.dtCz^-Zg) - w.Q.dt.h' = wQdt (vg-vj) 

2g 



or 



or 



2g w J 



- h' = ;]^ + P2 + Z2 

2g w 



h' = (]^+Pl+Zl) -(I2'*"P2+Z2) 
2g w 2g w 



(14) 
(15) 



Here in (15) h' is seen to be the difference between the totals of 
the "heads" at the two sections. We may then call it the "lost 
head" due to friction. (Sometimes spoken of as "friction head.") 
Again equation (15) may be thus expressed in words, - 

In steady flow, with friction , the total head at an y 
section is equal to that at any subsequent section plus the lost 
head occurring between sections . 

The study of causes or conditions leading to loss of head, 
and the methods for calculating their proportions form a consider- 
able portion of Hydraulics and will be discussed in later chapters. 

22. Orifices with Sharp Edges .- ?«'ater flowing through an 
orifice having sharp edges, presents the following characteristics. 
As it leaves the orifice it gradually contracts to form a jet whose 



cross-sectional area is 
(Fig. 25) This is due 




/^. ^s: 



somewhat less than that of the orifice. 

to the motion the particles have along the 
wall of the containing vessel before com- 
ing to the orifice. The contraction is 
not complete until the plane a-b is 
reached. For circular orifices having 
their edge in the same plane as the inner 
wall of the vessel, a-b is distant from 
that plane approximately one half the 
diara. of the orifice. At this point in 
the jet the pressure becomes atmospheric 
due to the surrounding medium. In the 
short preceding portion the pressure can 
be easily proved to be greater than at- 
mospheric . 

Let Fig. 26 represent such an 
orifice in the side of a large tank or 



-23- 



reservoir, having a depth of water, or 
head, on its center equal to h. This 
head is maintained constant by the inflow 
at E, and we will assume that the reser- 
voir surface A-B is very large compared 
with the area of the orifice. This means 
that the particles in the surface have no 
downward " velocity of approach. " Elimi- 
nating, for the moment, all friction, we 
will apply Bernoulli's Theorem to a parti- 
cle in the surface A-B and in the contract- 
ed section at C. For a particle in A-B 
the velocity-head is zero as the vel. is 

zero. The pressure must be that of the atmosphere which surroiinds 
it, giving Pa/w as the pressure-head. Lastly, if we assume our 
datum plane to pass through the center of the orifice, the potential 
head is h. At C the vel. head is v2/2g, v being for the present 
unknown. We have seen that the pressure is atmospheric and the 
potential head la zero. There then results, - 

r2 




f/^26. 



+ P& 
w 



h = II + Pa . 
2g 1? ^ " 



or v = ¥2gh 



which is the theoretic value of the velocity in the jet. Sever- 
al important deductions at once follow. We see that the water 
attains a velocity equal to that which it would have if it had 
fallen freely through the height h. In corroboration of this 
fact we would expect that, if the orifice were horizontal and the 
jet directed upward (Fig. 27), the stream would rise to a height 

equal to the head which produced it. 
Experiment verifies this, although for 
heads over 6 or 8 feet the resistance 
of the atmosphere becomes great enough 
to cause a slight discrepancy. This 
is more marked as the head increases. 
Prom the relation h = v2/2g it will now 
be seen why in Art. 20 we gave the name 
velocity-head to the term v2/2g,- it 
being the static head which would pro- 
duce the velocity v. This relation 
holds good for all cases of steady flow 
without friction, as we shall see later. 




25. Coefficient of Velocit y.- It is found by experiment 
that the actual value of v for a jet formed free in. air is a lit- 



-24- 

tle less than that found from v =■ ^2gh, mostly because of friction 
at the edge of the orifice but also because water is not a perfect 
fluid. The loss in velocity is about 2 per cent., giving as a 
value for actual velocity 



Actual Vel . = .98^ 2gh = c^\| 2gh 



(16) 



where .98 is called the Coefficient of Velocit y. The latter 
varies but very little in a range of several hundred feet of head, 
so we may use the value .98 in all numerical work. 

It might be noticed that the theoretic value of v ap- 
plies strictly only to orifices in a horizontal plane, all parts 
of the orifice being under the same head. With the orifice verti- 
cal and h measured to its center, the value of v obtained from 
V = ^ 2gh will not be a mean v alue for the cross-section of the jet, 
inasmuch as v varies .with ^'h. The difference, however, is but a 
small fraction of one per cent, and is cared for by the coefficient, 



24. Coefficient of Contraction .- This may be stated as 
the ratio of the area of the contracted section to that of the 
orifice. It varies with the size of the ori- 
fice and with the head, having a mean value of 
about 0.62 



a =0.62 
a 



25. Coefficient of Discharge .- The 
quantity Q flowing from the orifice per second 
may be stated as the product of a* and the ac- 
tual vel. past that section, and we may write 



Q = a'v' = aCgV' = aCQC^^2gh 
or Q = c^a^2gh 



(17) 




shall call it the 



where a^2gh represents the theoretical dis- 
charge and C(j is the product of c x c . \ 
C oefficient of Discharge and its mean value will be 0.62 x 0.98 = 
0.61. Since it varies with Cq and c^, for strictly accurate work 
its value must be determined for each particular case. Table II 
gives values under varying heads and sizes of orifice, having been 
determined experimentally from a large number of observations. 
It is part of a more extended table given by Hamilton Smith, Jr. 
in his "Hydraulics." 

86. Large Vertical Orifices tender Low Heads . - So far we 
have dealt with an orifice whose vertical dimension has been small 



-25- 



compared v/ith the head upon it. With the head low arid the orifice 
large, it will be foLind that the velocities of water particles in 
a vertical plane through the jet, differ considerably as they are 
under different heads . It will be shown, however, that this con- 
dition of things does not materiall y affect our niimerical results, 
provided the head is at least twice the vertical dimension of the 
orifice. 

Case I. Circle. - As before, h will rep- 
resent the head on the center of the 
orifice. If A-B be any elementary 
strip, drawn horizontally across the 
orifice at a distance x from its center, 
we have for the small discharge through 
it, 



Si/r/i/ce 




/yy^S. 



dQ = dA.v = 2\|r2-x^.dx x \|2g(h-x) 
= 2^2^ (r2-x2)V2(i,_j,)l/2^ 

By making x to vary between the values 
-r and +r, the integration of this ex- 
It will 



pression will give the discharge from the entire orifice 

be necessary to expand the term (h-x)V2 ^^ ^j^^ binomial theorem. 

(h-x)V2 ^ ^1/8 . ^-1/2^ _ h-V2^S . t,-5/2^5 . ^^c... 
.2 8 16 

.-.dQ = 2V2i [( r^-x^ ) V^h VS- ( rS-xS ) V^x - ( rS-x^ ) V^x^ - ( r^-xS ) ^/^x^ 



2h 



1/2 



8hV2 



16h 



5/2 



- etc . . . . dx 
Each term of this is now possible of integration ajid there results 
Q a-n'r2\(2gh (1 - r^ - Sr"^ - 10 5 r^ - etc ) (18) 



32h2 



1024h^ 



65537h 



which is an exact theoretic formula for the discharge. An in- 
spection of the parenthesis quantity shows it to have a value less 
than unity, and the discharge is therefore less than that given by 

the formula 

Q = af2gh 

previously obtained for relatively large heads. This is what we 
might have anticipated. 

Let us inquire into the numerical value of the parenthe- 
sis quantity as we assign different values to the ratio h. When 

r 



-26- 



h/r = 2, or the iiead is just equal to the dieimeter, the quantity 
becomes 0.992. With h/r =4, or h = 2d, the value is 0.998. 
This latter figure shows that with all ordinary ranges of head we 
may neglect the refinement of Formula (18) and for figuring actual 
discharge use 



Q = cafigh 
the value for c being foiond in Table II 




Case II. Rectaiigle .- In this case the 
small discharge dQ through an elemen- 
tary strip parallel to the surface may 
be written, as before. 



or 



L_>,^ ^^^^ 



dQ = dA.v = b!dx.^2g(h-x) 



dQ = bj2g (h-x)-'-/^.dx 



.where the limits of x are -d/2 and 
+d/2 . If we expand (h-x)-'-/" as before 



Q 



b^f2ihr' 



or 



d/i 
-d/2 

Q = bd\j"2ih (1- 



(1 - 



_x 
2h 



x"^ 



- 5x^ 



- etc. 



Sh*^ 



16h" 



128h^ 



)dx 



- etc.) 



(19) 



QSh' 



2048 ^4 



As in the previous case, we see that the value of the parenthesis 
is less than unity. If h = d, its value becomes 0.989 while for 
h = 2d, it becomes 0.997. Then we may say that for heads greater 
than twice the depth of the orifice, we may figure the actual dis- 
charge from 

Q = ca\j2gh 

Values of c for sjq uare orifices are found in Table III. Table IV 
gives values for c for rectan gular orifices 1 foot wide and of vary- 
ing depths. Both tables are taken from Hamilton Smith's "Hydrau- 
lics." The experiments of Poncelet and Lesbros showed that with 
rectangular orifices the coefficient varies with the smaller dimen- 
sion and is unaffected by the larger dimension if it does not ex- 
ceed 20 times the smaller. 



-27- 



27. Reca p itulation .- It is well to bear in mind the con- 
ditions under which we have so far studied orifices. We have as- 
sumed, - 

(a) Ratio of reservoir surface to orifice area very large, 

i.e. No velocit y of a pp roach . 
(h) Reservoir surface and jet both under atmospheric 
pressure. 

(c) A shar p edged orifice. 

(d) Kg suppression of the contraction. 

A departure from any one of thene conditions will lead to material 
changes in tho flow, the natures of which will be shown in the suc- 
ceeding paragraphs. 



28. Velocity of Approach. - Figure 31 shows, in longitud- 
inal section, a channel bringing water to an orifice. Because of 

a relatively small cross-sec- 
tion, there exists in the 
channel a vel. of approach. 
Let us call this V and A be 
the area of the cross-section. 
Applying our theorem to the 
points m and n we obtain 

v2+Pa+h=v2^Pa 
2g w 2g w 

or 




/^,J/ V^ + Pa 



XI + ISl + 



this being the theoretic velocity, 



V = \l2gh + V^ 

Since flow is steady 



(20) 



AV = cav 
Substituting in (20) aind simplifying, 



and V = cav 
A 



V = 



2£h. 



l-/ac>' 
^ A' 



For actual vel, 



V. = c. 



2gh 



and 



Q = c^acY 



ggh 



A 



V-'T ^(i)'-(f) 



2Eh_ 



^P 



(21) 



In this equation it will be noticed that V does not appear. 

Hence it is convenient to use in numerical problems where V is not 

known but the cross-section A is given. 



-28- 



(1) 



29. Problems. - 

Given.- h = 4 ft. 

a = 4 sq.in. 
A = 16 sq.ft 
c = 0.60 



Q = a 



2£h. 



^(^)^-(!)^ 



4 
144 



64.4 X 4 



.0278 



\| . 36 "^ 16 
Q = 0.27 c.f .p.s. 



Note. - Had Q been calculated v/ithout reference to the vel. of ap- 
proach, the error would have been lees than one per cent, and, in 
general the error will be negligible if the area of cross-section 
be at least 15 times the area of the orfice. This is easily 
proved by assuming values of a/ A in our general formula and com- 
paring resulting Q's. 

30. Problem. - An orfice of l/2 sq. inch area is located in 
the end of the closed channel A,B. (Fig. 32) An open tube C-D is 

tapped into the pipe as shown, and into 
this the water rises to a height of 4 
feet above the orifice. Assuming c at 
0.60 and that the water in the pipe has 
a velocity of 1 ft. per. sec, find the 
discharge. Since the pressure at any 
point is measured by its depth below 
the free surface D, the pressure head 
at m is 4 ft., m being in the horizon- 
tal plane of the orifice. Writing Ber- 
noulli's Theorem for points m and n 




f/^JZ 



64.4 



+ 4 + = -JL 



64.4 



+ 0+0 



V = 16.1 f.p.s. 

Q = 0.60 X .5 X 16.1 = .034 c.f. p.s. Ans . 
144 

31. Problem .- Assuming that in the previous problem the 
open tube is replaced by an ordinary pressure gage, find the dis- 
charge when the gage registers 40 lbs. p.s. in. 



-29- 



52. Flow Linder Pressure. - If either reservoir surface 
or jet be under pressure other than atmospheric, the formula 
Q = caj2gh does not directly apply. However, as it is merely 
a change in pressure conditions at these points, the application 
of the general theorem will tmffice for solving. 

For example, Fig. 33 shows in section a ves- 
sel partly filled with water aiid fitted 
with an air tight piston. A force of 1000 
lbs. being applied, it is desired to find 
the velocity of flow from A at the moment 
when the piston is in position shewn. As- 
sume Cy = .98 and area of piston = 1 sq.ft. 
For points m and n we have, (neglecting 
vel. of approach) 




/y^.J3 



+ 1000 + 2 = 



-1-0+0 



62.5 



64.4 



V = 34,3 



Actual V = .98 X 34.3 = 33.6 f.p.s, 



Ans , 



35. Problem. - 'i/Jhat will the 
velocity of flow from the orifice in 
the side of the tank (Fig. 34) when steam 
under pressure of 120 lbs. p. s. in. fills 
the spece above the water, and the re- 
ceiving tank has in it a pressure 4 lbs. 
less than atmospheric? Again Bernoulli's 
Theorem for points ra and n gives 



-t- 134.7x144 -(- 6 = 



62.5 



Actual V = 



64.4 
V = 137 



+ 10.7x144 -I- 
■ 62.5 



X 137 = 134 f.p'.i 



/20*p.s.//?. 




/y^. 34- 



34. Submerged Orifice.- If an orifice discharge wholly tin- 
der water it is said to be "submerged." The marked characteris- 
tic of this flov.' is the entire suppressior} of the contraction and 
cons equently a change in coefficient. That the velocity equals 
^J2gh where h is the difference in the water levels, can be easily 
shown, assuming atmospheric pressure only to be on the two surfaces, 
For the points m and n we have 



+ 34 -I- h3_ = 



= v2 



-1- (34-)-hp) + 



2g 



-30- 



froiij which 




/y-^.3S 



r2 ^ 



v^ = 2g (hi-hg) 



or 



V = \j2gh Q.E.D. 

Here h in the " effective" head. 

Table V gives valuer for the 
discharge coefficient under varying 
sizes of orifice and effective heads. 
It is taken fronj Merrimam's Hydrau- 
lics and is based on experiments by 
Hamilton Smith, Jr. 



I 



I 

(7 



. 55. Standard Orifice .- Anything that tends to decrease 
the contraction beyond the orifice will cause an increase in the 
amoiant discharged per 
linit time. So far 
we have been dealing 
with an orifice hav- 
ing sharp edges so 
that the water in 
passing touches only 
a line. The term 
"standard" has been 
applied to such an 
orifice. Pig. S6 
shows various types 
of orifices of the 
same diam. The 
first (a) is the a- 
bove mentioned stand- 
ard orifice. The 
second (b) is a_ 

cloan-cut hole in a plate of measurable thickness. It has the 
same coefficients as (a) provided its thickness is small. The 
third (c) is the reverse of (a) and, while it forms a contracted- 
jet, its coefficient of discharge is much greater and depends on 
the shape of the bevels. The last (d) shows the inner edge care- 
fully rounded to conform to the shape of the contracted vein. 
The probable coefficient of discha.rge is near to ujiity. 

Since the coefficients vary from. about 0.60 to 1.0 de- 
pending on the shape of the orifice, it is very essential to use 
only stsjidard orifices when endeavoring to accurately measure dis- 
charge . 




f/^.S6 



1 



-31- 



56. Suppression cf the Contraction. - The location of the 
orifice with respect to its distance from the sides and bottom of 

the reservoir is also a matter of impor- 
tance. If it be flush with any one side, 
as in Fig. 37, the contraction on that 
side will be wholly suppressed. Even 
moving it away slightly will not restore 
the contraction, and it is not until the 
clearance is made at least three times 
the smaller dimension of the orifice that 
the contraction becomes complete. There 
must be a, free lateral approach to the or- 
. if ice from all sides, existing for a dis- 
tance equal to three times the least di- 
mension . 



/7y.J?/ 



Orifices are sometimes con- 
structed as in Fig.38(a) and the 
plate A-B prevents this lateral ap- 
proach. A preferable construction 
would be that shown in Fig. 38(b). 



orifices 



57 . V.'ater Measurement by 
, - The orifice furnishes 



the most accurate way of measuring 
moderate volumes. If all the precau- 
tions enumerated in the preceding ar- 
ticles be observed the results should 
not be -in error much over 1 per cent. 
Of course the head must be kept very 
steady but this can generally be done 
automatically . 





/7^.d8 




/y^.J3. 



58. Discharge under Dropping Head .- It is 
often necessary to find the time required 
to, empty a reservoir or to dravv d.owa the 
surface level a certain amovmt . 

Let h]_ be the head on the orifice at 
the moment of opening and Yi^, "^^^ head at ' 
the end of some interval of time, _t. We 
will assume the horizontal cross-section 
of the reservoir to be constant. At any 
particular instant the vel. of flow is 

v = \l2gh 



-32- 



where h is the head at that instant. The quantity that would be 
discharged per second under that head is 

Q = ca\|2gh 

and in a very small fraction of time, dt, a small quantity dQ = 
ca 2gh dt would be discharged. In the same small interval of 
time the head would drop a small amount dh and the amount passing 
from the reservoir can bo expressed as the drop times the cross- 
sectional area, A. 

i.e. dQ = dh X A 
Equating these values, of dQ, 

dh X A = oa\J2gh dt. 
and 

A.dh 



dt = 



caf2gh 



If h be made to vary between h^ and hg the integration of the 
above will give 

t = ^ 



ca^^g" 



1-1/2 



h-'-z^dh 



^hg 



or 



t = ^-(<^-^) (SS) 



which is the time in seconds necessary to drop the level from hi 
to hg . 



-33- 



CHAPTER II 



Flow through Mouthpieces. 

59. General. - If in any manner a mouthpiece be added to an 
orifice, we may expect changes in both jet velocity and quantity 
discharged. As for velocity, we notice a general diminution on 
account of increased frictional resistances, eddyings, impact, etc. 
The quantity discharged is either increased or diminished accord- 
ing to the construction of the mouthpiece. 

40. Standard Short Tube. - If a short cylindrical tube, 
having a length of 2 to 2-1/2' diameters, be attached to the ori- 
fice on its outer side, vie have what is known as the standard 
short tube. (Fig. 40) If the flow be started by first stopping 

the tube at exit and then al- 
lowing the water to escape, the 
jet will issue in parallel fila- 
ments, suffering no contraction 
and filling the tube full bore. 
Inside the tube at m the jet is 
contracted by passing the sharp 
edges of the opening, but quick- 
ly expands to fill the tube. 
By reason of the eddying which 
follows the expansion, consid- 
erable friction is developed. 
This added to friction at en- 
trance and along the sides of 
the tube tends to materially 
reduce the velocity. If Bernoulli's Theorem be applied between 
a point in the reservoir surface and one in the free jet, we ob- 
tain, as we did for the orifice 




while from actual experiment V/eisbach foiAnd 

'~jh (23) 



V = .815 \|2gh 

This latter value would give for quantity discharged 

Q = .815a\|2gh 

inasmuch as the contraction coefficient is unity. Thus we have a 
discharge which is 1/3 greater than that obtained from the stand- 
ard orifice although the velocit y of the discharge is much less. 

If a glass tube be tapped into the mouthpiece at the 
point Where the contraction takes place and its lower end be im- 



-34- 



raersed in water (Fig. 41), the 
ing that the pressure in the c 




latter will rise in the tube, show- 
ontracted section is less than at- 
mospheric. It will be inter- 
esting to determine- the rela- 
tion between the height of the 
water column h-^ and the static 
head h on the mouthpiece. 

Between a point in 
the reservoir surface and one 
at ra we may write 



fi^.4/. 



+ p 



+ 
a 

w 


h 


— 


2g 


Pm 


+ 


lOE 


t 


head 





+ 



may be found if we make the as 
traction at m is 0.63 the same 
case, with steady flow, we hav 



. and if Vm and the Lost Head can 
be expressed in terms of h this 
genera] equation may be solved 
for Pja/w in terras of h. 

The value of v^2g 
sumption that the coefficient of con- 
as for a standard orifice. In this 
e 



Vjjj X .62a = v X a 



or 



from which 



= 1.59v = 1.59 X .815^2gh = 1.29 ^ 2gh 



Vm = 1.67h 
2g 

We have next to determine the value of the head lost in 
friction as the water enters the tube past the edge A-B, this be- 
ing the only sensible loss occurring between the points chosen. 
This is best done by considering for a moment the case of the 
standard orfice and the similar loss occurring there. The ac- 
tual velocity in the jet from the orifice was 



V = . 98\|2gh 



from which 



2g 



= .96h 



-35- 



showing that .04h was head lost. If we use for h its value as 
determined from this laGt equation, we may write 

Lost head = .04(^;g| "" ig^ ^ •^'^^ ^ 

and the loss is expressed in terras of the velocity-head found in 
the issuing jet. 

Returning to the short tube it seeraa reasonable to be- 
lieve that the loss at the edge A-B will be exactly .042 x the 
velocity head in the contracted. section, or 

Lost Head = .042 x 1.67h = .07h (24) 

Substituting in the general equation the values now 
found for v^2g and Lost Head, and remembering that Pg/w = 34 ft., 
we obtain 



+ 34 + h = 1.67h + Pm ^ 



07h 
w 



or 



-^ = 34 - .74h (25) 

w ^ ' 

That is to say, the pressure-head at the contracted section is 
less than atmospheric by an amount equal to 3/4h. This latter 
is the height of the water in the glass tube and we have now the 
relation 

hi = |h 

The correctness of this relation and our assumptions was 
demonstrated by Venturi in a set of beautiful experiments. With 
a head of 0.88 meters he obtained a water column of 0.65 meters 
which is an exact verification. 

Referring to Fig. 41 it will be seen that if the dis- 
tance B-C is' less than h^ , the water will enter from the glass 
tube into the mouthpiece and be expelled with the jet. This is 
the principle of the "jet pump." 

Furthermore, as the height of the water column cannot 
exceed 34 ft: (approx.) it follows that the pressure in the con- 
tracted section will be absolute zero when the static head is 
about 45 ft. (Since 34 = 3/4h) Experiments conducted in the Hy- 
draulic Laboratory of the J/lass. Inst, of Technology in 1900 showed 
that when the head approached or exceeded about 42 ft. the flow 
became "troubled and pulsatory." This was probably due to the 
"breaking down" of the contracted section when all pressure was 



-36- 



reraoved. 

At Cornell Univ. some experiments were made on tubes 

which had been made of such a form internally as to conform to the 

shape of the contraction. (Fig. 42) It was found that the flow 

was increased about lOfo in these tubes, 
which may be explained by saying that the 
eddying would be diminished and some loss 
by friction prevented. 

The short tube is of practical 
importance only as it serves later to aid 
in the study of flow through long pipes. 
As a device for measuring water it is much 
less reliable than the orfice since lit- 
tle is known about its coefficient as the 
length of tube and height' of head vary. 

The value 0.815 may be considered only a mean taken from many ex- 

■neriments . 




/^.^^. 



speci 
engin 



41. Conical Converging Tube. - This mouthpiece is of 
al interest as it approximates the nozzle used for fire and 
eering purposes. It consists of a frustrum of a hollow cone 

with the larger end placed at the 
orifice. (Fig. 43) From such a 
tube the water flows in a jet hav- 
ing a slightly contracted section 
(a-a) just beyond the tip. The 
amount of contraction being very 
slight, the contraction coefficient 
is not far from unity and v;ill in- 
crease as the angle of convergence, 
B, decreases. The velocity coeffi- 
cient, on the other hand, is found 
to diminish as B decreases, and has 
its maximum value of 0.98 when B = 
180° The tube under this condition 
becomes an orifice. 

These variations are well 
shown in the accompanying table, 
based on the work of D'Aubisson and 




Castel who experimented with this type of mouthpiece; 



Angle B. 

Crr 



0"0 



3" 10' 7°52' 10° 20' 13°24' 19 28' 29 58' 



48''50' 



0.829 0.894 0.931 '0,950 0.962 0.970 0.975 0.984 
1.000 1,001 0.998 0.987 0.983 0.953 0.919 0.861 
0.829 0.895 0.929 0.938 0.946 0.924 . 0.896 0.847 



The coefficients of discharge were obtained by direct 
measurement of the volumes of water discharged and the velocity 
coefficients were calculated by measuring the path of the jet. 



-r37- 



'V 



The contraction coefficients were then deduced from 0^ = 0-5-0^ 

The method of obtaining the coeff . of velocity makes no 
allowance for the retardation of the particles in the Jet by the 
atmosphere; hence the coefficienta may be a little small and the 
contraction coefficients would then be too large as deduced from 
the above equation. This would seem to accovmt for the value of 
1.001 as seen in the second column. 

The table would indicate that the greatest discharge 
occurred when the angle of convergence was near 13° and, as the 
corresponding velocity was relatively high, this angle should give 
a very efficient nozzle. 

42. Tlozzles for Practical Purposes .- A nozzle is a modi- 
fied form of the above tube, shaped to do away with the contrac- 
tion in the free jet. This is accomplished sometimes by the ad- 
dition of a cylindrical tip as shown in Pig. 44(a). 




(Cf) 



Fi'^.'4-4: 




The contraction being eliminated there follows an increased dis- 
charge. (The velocity is not perceptibly changed.) If the tip 
be of short length the jet will pass througn without wetting it. 
Sometimes the whole nozzle is made convex inside so as to give a 
parallel motion to the particles as they leave the tip. (Fig. 44 
(b) ) . Either device results in a contraction coefficient of 
linity . 

For all classes of convergent tubes and nozzles we may 
write for actual velocity of jet, 



V = c 



v\|2gh 



and for quantity discharged 



ca 



■NJigh 



The value of the coefficients must be determined for every partic- 
ular nozzle . 

Nozzles are generally used on the ends oi pipe lines 
where there usually exists a conpiderable vel . of ai")proaoh. In 
this case, as in that of the standard orifice, it may be proved 
that actual vel. of jet is 



-38- 



V = Cy\ 2gh 



\ 2j 

\ l-,acx2 



and quantity discharged i's 



Q, = a 



l^ 



1 2 a ' 



(Art. 10) 



Here a would represent the area of the nozzle tip and A the area 
of the pipe. Since 

a _ d^ 
A ^2 ' 
. ■ D 

d and D being the respective diameters, we may write 



V = c. 



'I l-C^f^ 



^P 



(26) 



and 



Q, = a 



8gh 



<^)^- 



(-) 



:27) 



Of course if the velocity of approach be known in value, the di- 
rect use of Bernoulli's Theorem between a point in the pipe and 
one in the jet will give a theoretical value for v which may- be 
corrected by the coefficient. 

A very extensive study of nozzles and their coefficients 
was made in 1888 by Jolin R. Freeman and' his results may be found 
in the Transactions of the Amer. Soc. of Civil Engineers, Vol. XXI, 
Page 303 . 

45. Diverging Tubes.- It has been pointed out that the 
chief loss of head in the standard short tube is occasioned by the 
sudden 'enlargement of the contracted vein to fill the. full section 
of the tube-. If the interior of such a tube should be shaped as 
H ■ in Pig. 45 so as to not only fit the 

1 form of the contracted vein but also 

^ '- ,,,..... ^^^_=-«= ^^Q cause a gradua.l enlargement, this 

loss should be done away with and the 
discharge increased. Venturi and 
Eytelwein used such tubes but also 
/ya.4S^. rounded the inner edge to reduce fric- 



^1^^^SS2ZZZ^^^ 




-39- 



tion loss to a minimum. Their tube was 8 inches long, 1 inch 
diajn. at the smallest section and 1.8 inches diam. at the large 
end. The angle of flare was 5°9'. (Fig. 46) 
They obtained in their experiments a 
discharge nearly 2-1/2 times as great 
as though it had occurred through a 
standard orfice of the same diam. as 
the small section, and almost twice 
what would have been discharged by a 
standard short tube of the same diam. 
The velocity past m was much greater 
than that due to the head it was un- 
der and the pressure at ra was much 
less than an atmosphere. This last 
may be proved by writing Bernoulli's 
Theorem between m and n. The two 
points being on the same level and 
the velocity head at ra greater than 
at n, the pressure head at m must be 

smaller than at n. The increased discharge through the small 
section is therefore due to the diminished pressure upon it and if 
by any means atmospheric pressure were maintained there, the phe- 
nomenon of increased flow would be lost. Venturi proved this by 
boring holes through the sides of the tube at this point. 

If the flare of the tube be considerable or the head 
upon it low, some trouble may be had to keep it flowing full at 
the outer end. To remedy this it is sometimes made to discharge 
under water as in Fig. 47. If the head-on the outer end be hg 

then the effective head will be 




f/;^^6. 




f/^47 



h = hi - h2 

just as for the submerged ori- 
fice. 

The principle of the . 
flaring tube fir^ds its appli- 
cation in the "diffuser", which 
is used with many types of tur- 
bine water wheels to reduce the 
pressure at the outlet of the 
wheel.- (Church's Motors, p. 126.) 



-40- 



OHAPTER III 



WEIRS 



44. Definitions .- In general terms a weir Ie a notch 
cut in the upper edge of a vertical wall, through which water is 
allowed to flow for purposes of measurement. The common form of 




f/^.4S 



r/^.49. 



notch is a rectangle having its lower edge or "Crest" horizontal. 
Figure 48 shows a " Contracted Weir " so-called because the issuing 
stream is contracted at the ends. In contradistinction to this 
type is the "Sup pressed Weir ," (Fig. 49) whose end contractions are 
suppressed by making the sides of the notch flush with the sides 
of the feeding reservoir. The contraction is said to be "com- 
plete" and "incomplete" in the two cases respectively. In both 
v/eirs the surface of the water in the reservoir immediately back 
of the weir does not stand level, but assumes a curve as in Fig. 

50. 

The "Head" on a weir 
is not the depth of water ver- 
tically over the crest, but is 
measured, as shown, from the 
level of the crest up to the 
general reservoir surface. 




^j,,,,j,/,i//f///j^^^//^j/^r77y. 



. 45. Fundamental Equations 

^^^- ^O. for Flow .- If a large vertical 

orifice discharges with a head 
on its upper edge equal to zero, we have results analagous to the 
flow over a weir. Let Fig. 51 represent in a section a large 
orifice in the end of a reservoir or channel, the latter being so 
shaped as to produce a velocity of approach, c. 

Let V represent the velocity of discharge through any 
horizontal elementary strip at a distance x below the surface. 



-41- 



m 






T 

X 









_c__^ — 



A 



i» 



/7>.5/ 



For this small strip the discharge per linit time is 

dQ = dA . V = b . dx . V 

and for the entire orifice 

Q = bj V . dx 

To find V in terms of x write Bernoulli's Theorem between the 
points m and n giving 

c^+Pa+x=v^+p^+0 
2g "li? 2g -^i 
from which 

V = \j2g (x + h)-"-/^ 

where h = c2/2g. Substituting this value for v in the above ex- 
pression for Q there results 

Q = b\f2g J (x + h)-'-/^dx 

The quantity (x + h) may be regarded as a variable with limits 
(h2+h) and (hi+h) since d(x+h) = dx. With these limits the above 
equation becomes on integrating 

Q = 2 bjii" |(h2+h)^/2_(^^+j^)3/2J 

If h.1 be made zero our orifice becomes a weir and, by substituting 
H for h2 we obtain ^ 

Q = 2 b\[2^ [(H+h)^/^ - h^/^J (28) 



If there be no velocity of approach, h = and 



-42- 

Q = 2 bJig.H^/^ (29) 

3 

These two equations will give the theoretical discharge for either 
a contracted or a suppressed weir and may be regarded as fiinda- 
mental equations. The second of the two will give the discharge 
only when there is no velocity at the point where H is measured. 
This can only happen when the area of the cross-section of the 
channel of approach is large compared with the weir opening. 
This would be so if the weir were in the side of a pond or large 
reservoir. To apply either equation in practice it is necessary 
that a coefficient be introduced to care for frictional disturban- 
ces and contractions at the weir edges. The difference between 
the theoretical and actual discharges is quite large, so that we 
have to depend for accuracy upon the value of the coefficients. 
To ascertain these values under different conditions of head, 
length and type of weir has been the aim of many experimenters. 
They have tried also, to deduce new formulae with an aim toward 
simplicity and wide range of application. There can be no ob- 
jection raised to these formulae provided they are rational and 
are applied only when the conditions ixnder which they are used fall 
within the limits of the observations by which they and the corre- 
sponding coefficients were obtained. The value of h in the fun- 
damental equation we have seen to be c2/2g, c being the velocity of 
approach. Since in general this can not be measured directly and 
the corresponding value for h be determined, we have to use a 
method of trial in computing the discharge. To make this clear 
let us imagine the velocity of approach to exist .in a certain weir 
channel and that the coefficients for formulae (28) and (29) are 
known. 

The approximate quantity dischargedmay be first calcu- 
lated by (29), neglecting the velocity of approach. This latter 
may, in turn, be approximated from 

c = Q 
A 

£ being the average velocity in the channel, and A the area of its 
cross-section. If a value of h be determined thus, equation (28) 
will give a new value for Q. This will be nearer to the true Q, 
although still somewhat approximate. By using this value and 
again finding c and h, the formula may be used' a second time and 
a value of Q obtained which will probably be so close to the true 
one that the recalculation of the h would fail to produce a mate- 
rial difference in the result. 

46. Francis* Formulae. - In 1851, at Lowell, Mass., Mr 
James B. Francis carried on an extensive series of experiments with 
large sized rectangular weirs. Most of the weirs were 10 feet 
wide and the heads ranged from 0.'6 to 1.6 feet. As the result of 



-43- 



many experiments he proposed the two following equations. 

Q = 0.622 X 2 (b - _J, nH) pg" [(H + h)^/^ - h^/^1 (30) 
3 10 



eind 

Q = 0.622 X 2 (b - _1 nH) jig" H^/^ (31) 

3 10 

These may be regarded as identical with our fundamental formulae 
(28) and (29) save that he regarded the effective width of the 
weir to be less than b by an amount nH/lO due to the end contrac- 
tions. The quantity 0.622 is the value of the coefficient c, and 
n varies in value as follows,- ■• 

n = 2 when there are two complete end contractions, 
n = 1 when one end contraction is entirely suppressed, 
n = when both end contractions are suppressed. 

His equations may therefore be written, - 

(a) Contracted Weir .- /„ „/^ (32) 
Q = 3.33(b - _2 H) [{E + h)"^/"^ - h"^/ with vel. of approach. 

10 

Q = 3.33(b - _2 H) ^'^ no Vel. of approach. (33) 

10 

(b) Sup pressed Wain. ,/^-| 

Q = 3.33 b [(H + h)3/S _ h"^/ ^ with vel. of approach (34) 

Q = 3.33 b H"^/^ with no vel. of approach. (35) 

Francis' formulae have been very widely used, because of their 
simplicity, since they require no new value for the coefficient 
with varying conditions. Prom over 80 experiments he deduced a 
mean value for c of 0.622 and, believing that the contraction 
varied with H, introduced the term (b - 2/ 10 H) as before explained. 
Since his mean value of 0.622 varied about 7>fo from the extremes, 
his formulae should give results to that degree of accuracy pro- 
vided they be not applied to weirs that vary much, m dimensions 
of width and head, from the weirs with which he experimented. 
Generally b should be greater than 3H and H should not be very 
small, say under 6 inches. 

The method of correcting for velocity of approach and 
calculating h is the same as outlined in the previous paragraph. 



-44- 



47. Fteley and Steams'. Formulae .- Those two engineers 
experimented at Framingham in 1877 and 1880, with results differ- 
ing slightly from those of Francis. Their experiments seemed to 
show that the amount of contraction did not depend wholly upon the 
head, but was quite irregular, varying also with the velocity of 
approach. For the case of the suppressed weir they proposed for- 
mulae as follows. 

When the velocity of approach is negligible. 

Q = 3.31 bH'^/^ + .007b (36) 

and when it is so large as to influence the discharge 

Q = 3.31 b (H + 1.5h) '" + .007b (37) 

The correction for the velocity of approach is made by adding 1.5h 
to the measured head, h being derived, as before, from h = ■c2/2g. 
To properly apply these formulae the weir should be from 5 to 20 
feet wide and the head should not be less than 0.07 feet. 

48. Bazin's Formula for Suppressed Weirs .- M. Bazin of 
France, in 1888, after a series of carefully conducted experiments 
upon weirs with end contractions suppressed , proposed the follow- 
ing formula for the discharge from such v/eirs. 



= 2 cfl + 0.55( H )^ 
3 L (P+H)2_ 



bH\|2gH (33) 



where c has the value 



c = 0.6075 + 0.0148 
H 

and p represents the height of the weir crest above the bottom of 
the channel of approach. It should be noted that the formula 
provides for the influence of the velocity of approach and thus 
allows Q to be calculated directly without resorting to the method 
of trial. Bazin's weirs ranged from 0.50 to 2.00 metres in width 
and the heads from .05 to .60 metres. His formula applies there- 
fore to short weirs under large heads rather than to long weirs 
and \qvi heads . 

49. Formulae of Hamilton Smith. - In 1886 Hamilton Smith, 
a noted American hydraulic engineer, published in his "Hydraulics" 
a very careful resume and compilation, of the best and most trust- 
worthy experiments on weirs. From a study of the work of Francis, 
Fteley and Stearns, and others, he proposed for 



-45- 



Contracted Weirs 



,3/2 



Q. = c X 2 b\l2g K ' with no vel. of approach. 
3 

Q = c X 2 bv|2g (H + 1.4h)'^/^ with vel. of approach. 
3 
and for 

Suppressed Weirs 

Q = c X 2 b\|2g IT' with no vel. of approach. 
3 

I — 3/2 

Q = c X 2 b\j2g (H + 4 h) ' with vel. of approach. 

3 3 



(40) 
(41) 

(42) 
(43) 



These formulae differ from Francis in the method of al- 
lowing for effect of end contractions and for velocity' of approach. 
Formulae (40) 5ind (42) are identical in form with the fundamental 
equation (29), having values of c dependent on the type of weir 
Eind varying with its length and also with the total or effective 
head, (H + nh) . Formulae (41) and (43) vary from (28) in the 
maj^ner of correcting for velocity of approach. Again c depends 
on the type of weir and varies with b and (H + nh) . 

Tables VI and VI I give values for £ for a wide range of 
conditions. The first column in each table represents the effec- 
tive head (H + nil) but, in selecting a coefficient for use, no 
appreciable error will result if we use H instead, nh being very 
small. For heads and lengths not given, c may be found by inter- 
polation. 

50. Submerged Weir .- If the water level on the down- 
stream side of a weir be made to stand above the level of the 
crest, the weir is said to be "submerged." Using a weir with no 
end contractions, Fteley and Steams experimented under the above 




A 


. 


e 


n 




s^ 






f\ 


t 


< h i 


y ' 




-* * 





r/^.52. 



-46- 



conditions to determine a rational fonnula for flow and coeffi- 
cients for the same. Fteley and Steams deduced their formula 
by considering that the flow took place in two parts,- one through 
the weir ABCD (Fig. 52) and the other through the orifice CDEF. 
The theoretical discharge through the weir would be 

,3/2 



Q = 2 byf2i (H - h)' 
3 



and through the orifice 



Q = bh^ (H - h)^/^ 



By adding these and introducing the coefficient there would re- 
sult 



Q = c X 2 b\f2g (H + 1 h)(H - h)-""/^ 
3 2 

Pteley and Steams replaced c x 2J2g by m and proposed 

Q = mb{H + 1 h)(H - h)^^^ 
2 



(44) 
(45) 



m having a value dependent on h 
table . 



H as shown in the accompanying 



h - H = 
m = 


0.00 
3.33 


0.04 
3.35 


0.08 
3.37 


0.12 
3.35 


0.16 
3.32 


0.20 

3.28 


0.30 
3.21 


h 4- H = 
m = 


0.40 
3.15 


0.50 
3.11 


0.60 
3.09 


0.70 
3.09 


0.80 
3.12 


0.90 
3.19 


1.00 
3.33 



Experiments on submerged weirs are too few to give accuracy and 
certainty to the results obtained by formulae. Such weirs should 
therefore be avoided for accurate measurements. It may sometimes 
happen that, during the gaging of a natural stream, an unforseen 
rush of water may submerge the weir. The above formula will then 
aid in a close approximation to the discharge during the flood in- 
terval . 



51. Triangular Weir.- Triangular weirs have sometimes 

been used where the quantity of 
water flowing was not large. 
The customary arrangement is 
shown in Fig. 53, both sides of 
the notch being equally inclined 
Inasmuch as the issuing stream 
is of similar cross-section for 
all heads, the value of the co- 
efficient should be fairly con- 




-47- 



stant. Experiment has shown this to be the case. 

The formula for theoretical discharge may be obtained 
as follows. In Fig. 54 let x be 
the head on an elementary horizontal 
strip. From similar triangles its 
length is b(H - x) -^ H and for its 
area we have 



dA = b (H - x)dx 

H 




/y^.S4 



The discharge through the strip is 

dQ = b (H - x)dxJ2gx = bJSg (Hx-*-/^ - x^^^Jdx 
H H 

and if this be integrated with H and as the limits of x we obtain 

Q = _4 bJsi H^/^ (46) 

15 

The sides of the triangle being equally inclined, b = 2H tanqf. 

.'. Q = _8 tanevJSg H^/^ 

Making the vertex 90 so that o< = 45 we have, 

actual Q = c X _8^ ^'^ (47) 

15 
where c has a mean value of 0.592 when the rsmge in H is from 0.2 
to 0.8 feet. Using this value, 

,5/2 



Q = 2.53 H 



(48) 



If vel. of approach exists H should be replaced by (H + 1.4h) as 
in the case of the rectangular weir. 

52. Trapezoidal Weir of Clppoletti .- This weir is men- 
tioned more because of its ingenious design and common use in some 
sections of the country, them because of its value as an efficient 

measuring device. It has 
the advantage of a certain 
amount of constancy in its 
coefficient . As the name 
indicates, it has a notch of 
trapezoidal form as shown in 
Pig. 55. The side slopes 
are alike and generally have 
a slope of 1 in 4. The 




\ 



-48- 



reason for this is of interest. The discharge may be divided 
into tvTo parts,- one through the rectangular area of length b, 
and the other through a triangle having a base width of 2d. The 
total discharge is therefore greater than from a recteingular con- 
tracted weir of length b. Cippoletti proposed giving the sides 
such a slope that this increase would be just equal to the decrease 
in discharge, through a contracted weir, caused by end contrac- 
tions. This would make the trapezoidal weir the equivalent of a 
rectangular suppressed weir of length b. The increase, being 
the discharge through the end triajigles, may be written from 
equation (46) as 

Q = c X 4 X 2d\|2g.H^/^ 
15 
emd the decrease due to end contractions is, according to Francis, 

Q, = c X 2j2g'x 0.2H^/^ (See equation ^^.^ 

Equating these values and assuming the two values of c to be 
alike, there results 

d = H 
4 

giving the slope which Cippoletti recommended. From such a weir 
we may then figure Q by Francis' formula 

Q = 3.33bH^/^ 

Cippoletti believed, as a result of experiment, that the value 
3.33 was too small, and proposed 

Q = 3.367bH^^^ 

but, as experiments to confirm this are lacking, it is probable 
that Francis' formula will give just as reliable results. 

53. Weir Measurement.- Suppression of the Contraction. - 
Experiment has shown that the weir is a very satisfactory 
method of measuring large quantities of water, provided certain 
precautions be taken. 

(a) Measuring the Head. - We have seen that the head must 
be measured at a point some distance back from the crest to avoid 
the surface curve. The devices for measuring vary, but in precise 
work the hook-gage is commonly used. (Description given in lec- 
ture.) This device enables the head to be road to the nearest 
thousandth of a foot with ease, an accuracy generally beyond that 
of a coefficient. 



-49- 



(b) Suppreaaion of the Contraction. - Care must be taken 
in constructing a contracted weir to see that the notch is far 
enough away from the sides and bottom of the feeding canal to pre- 
vent interference with the complete contraction at the edges. 
Generally speaking, a clearance of 3 times the head should be had 
between the botfom of the canal and the crest, and between the 
aides of the notch and the sides of the canal . 

In all cases the space beneath the falling stream 
should be open to the air at the ends of the weir to avoid the 
existence of a partial vacuum under the sheet, a condition which 
would tend to increase the discharge. 

(c) Sharpness of the Crest. - As already pointed out, 
the coefficients in general use were obtained from experiments on 
weirs having sharp edges. It will be shown in the following ar- 
ticle that a departure from this condition leads to increased or 
decreased flow (according to conditions) and therefore to loncer- 

tainty . 

54. Weirs with Rounded or 
Wide Crests. - Messrs. Pteley 
and Steams experimented with 
weirs having the upstream edge 
of the crest slightly rounded. 
(Fig. 56) When th6 'rounding 
was made by a quarter circle 
having a radius of less than 
0.5 inch, they foxxnd that the 
discharge was increased and 
could be determined by the 
usual formulae, provided 0.7 
the measured head . 
but made so wide as to cause 




/7^.56 



the radius (in feet) be added to 

If the crest be squared 
the water to touch again 
after leaving the inner 
edge, the amount dis- 
charged is generally 
diminished by the in- 
crease in friction. 
Merrimam in his "Hy- 
draulics" illustrates 
two cases as follows. - 
pig. 57(a) shows the 
crest just wide enough 
to slightly interfere 
with the descending sheet. The 
creased according to whether the 
with air or is a partial vacuum, 
resemble Pig. 57(b) the flow will 
crease in frictional resistance. 




flow may be decreased or in- 
space over the crest be filled 
If the crest be so wide as to 
be diminished by the large in- 
Any of the above conditions 



-50- 



should be avoided in precise measurements. 

55. Dams used as Weirs. - The many dams existing on 
natural streams offer opportunities for gauging flov^ if only 
their coefficients may be ascertained. As the shapes or pro- 
files of dams vary greatly in detail it would be a difficult and 
laborious task to determine and tabulate the coefficients for the 
many various types. However, two excellent sets of ' such experi- 
ments on dams have been recently completed. The first was by 
Bazin in 1897, carried on at the request of the French Government. 
The second, in 1898, was under the direction of G. W. Rafter for 
the U. S. Deep Waterways Commission, carried on at the Cornell 
Hydraulic Laboratory. The results of both sets of experiments 
appear in a paper by Mr. Rafter in the Transactions of the Amer. 
Soc , of Civil Engineers, Vol. 44, 1900. The formula proposed is 
in the form 

Q = c X 2 JS^ . bH^/^ = MbH^/^ (49) 

and values of M are given for a great variety of profiles and con- 
ditions. 



.^' 



-51- 



CHAPTER IV 

Flow in Pipes 

56. General Equation for Flow. - If a straight friction- 
less pipe of indefinite length were attached to an opening in a 
reservoir (Fig. 58) and made to discharge under a constant head 

into the atmosphere, 
the velocity of flow 
at the outle t would 
be V = ^2gh. The 
existence of fric- 
tion however at the 
entrance to a pipe 
and along its sides 
materially reduces 
this value and, in 
this frictional resistance 
a feeble flow. The head h in 




f/^.58 



a very long pipe of small diameter, 

might be so large as to permit only _ 

this case would be almost wholly used up in overcoming friction. 
Between the points m and n we can write 



p„ + h = V^ + p„ + + Lost Head (Art. 21) 
a. « "■ 



2g -S? 



\ 



or 



h = v£ + Lost Head 
2g 



(50) 



The last term covers all the losses in head occurring between the 
reservoir and outlet, no matter how caused. If each of these can 
be expressed as a function of the actual velocity then equation 
(50) will contain but one unknown, v, and may be used as a general 
equation for solving problems of flow. 

57. Losses of Head in Pipes. - In a straight pipe of uni- 
form bore, the losses at entrance and by pipe friction are the only 
ones to consider; but if there exist bends and sudden changes in 
diameter, or if the flow be obstructed by partially closed valves, 
etc., then each of these conditions will result in a loss of head. 
A discusBion of such losses follows. 



58. Loss at Entrance. - The conditions of flow in the pipe 
at the point where it joins the reservoir are similar to those in a 
standard short tube which discharges against a head of water h]^ 
(Fig. 59) . This head, h, , may be likened to the friction head in 
the pipe, both being a resistance to flow. 



-52- 



In each case the difference h - hj^ is the head available for over- 
coming resistances and creating a velocity of flow. In Pig. 59 

we have, by Art. 38, 




V = 0.815 ^g(h-hi) 
from which 

~- = 0.66 (h-hi) 

so that 



Lost Head = 0.34(h-h3^)! 

0.34(^ - 0.66) = 

2g 



F/^.S9. 



0.52 v^ 

2g 
Prom our above comparison we may reason that this must represent 
the head lost at entremce in the case of a pipe, and write 



Lost Head at Entrance = 0. 5 v£ 

2g 



(51) 



59. Loss by Pipe Friction.- Of the losses mentioned in 
Art. 57 the chief in magnitude (unless the pipe be quite short) is 
that due to friction against the sides of the pipe. To accurately 
express this loss in an equation which will apply to pipes of all 
sizes having inner surfaces of varying degrees of roughness, and 
transmitting water at different velocities, has so far been beyond 
the ability of the hydraulician. Even temperature enters into the 
problem to add to its complexity. As a result of many experiments, 
however, it is known that the amount of resistance offered is 

(1) Independent of pressure in the pipe. 

(2) Varies directly with the extent of frictional surface. 

(3) Varies nearly with the square of the velocity for veloci- 

ties above what is known as the "critical velocity." 
Below the critical velocity the resistance varies with 
the first povver.- (See discussion of Critical Velocity.) 

Any formula, therefore, to properly express this loss must have a 
rational basis and recognize the above conditions. 

One treatment of the problem is as follows. The accom- 
panying sketch represents a portion of the pipe shown in Fig. 58. 
The reservoir and pipe are supposed to be filled with water and 
the outlet stopped. If the latter be opened the action of grav- 
ity will cause the water to flow with increasing velocity until 



-53- 



the resulting increase in friction between the water and pipe_, and 
among the water particles themselves^ causes a resistance that is 
capable of balancing the action of gravity. The accelerating 

force then becomes zero and 
steady flow ensues. In any 
cross-section of the pipe the 
mean velocity has then reached 
its maximum value. Across the 
section the velocity will vary 
from a maximum at the center to 
a minimum at the sides . We 
will neglect this variance, 
however, and consider all the 
particles as having the mean 
velocity v. Referring to 
Fig. 60,- m and n are two points on the axis of the pipe at dis- 
tances Zju and z-^ above the datum. Between the two points we may 
write, (Bernoulli's Theorem) 




%. 60. 



2S 



Pm 



+ Zjh = 



_ ^n 



Pn 



in + £n + 2 + Lost Head 
2g w " 



and since the two velocities are equal (the pipe being of uniform 
section) 



Lost Head = (5l + z^) - (hi + z^) 



(52) 



To otherwise express the loss in head, consider the cylinder of 
water between sections m and n, and the forces acting on it. 

(Fig. 61). Apm and Ap,^ are the 
pressures exerted on the ends by the 
adjacent pai^ticlee, A representing 
the area of cross-section. R is 
the total frictional resisting force, 
and Awl is the weight of the water. 
The motion being uniform we may write 

Apm + Awl . sineo< - R - APn = 
or 







£a + lsineo<- Pn = 



w 



w 



From Fig. 60 we have, Isineo^ = Zm - z- 
which inserted in the above gives 



'n 



-54- 



and this combined with (52) results in 

Lost Head = -r^ 
Aw 

If now the frictional resistance per -unit area of pipe surface be 
P so that R =TrdlF, we may write 

Lost Head = ^TTdlF = 4lF 

w.Trd2 ^d (53) 

The value of F we cannot measure, but assuming it t*o be sC function 
of v2 (v being above the critical velocity) so that P = cv2, we ob- 
tain 

Lost Head = — .c.i.yS 
w d 

or more simply 

Lost Head = f ^ v^ (54) 

where f represents 4c/w. It is important and convenient that f 
be a purely abstract number so that its value will be independent " 
of the system of units employed. Equation (54) may be thus in- 
terpreted. - 

Distance = f x distance x (distance)^ 
distance 



so that 



f = 



distance 



or f is a function of a distance. To make it an abstra^ct number 
there must be introduced into the right hand member a quantity in 
the form, l/distance. If we make this l/2g there will be secured 
the added advantage of having the lost head expressed in terms of 
the velocity head and we shall then have 

Lost Head = fi.f^ (55^ 

60. Determination of the Value of f .- Much experimental 
work has been done by Weisbach, Darcy, Prony, Fanning and others 
to determine values of f under the varying conditions which are 



-55- 



met with in practice. As a result the following facts may be 
stated. - 

(1) f decreases, in the same pipe, as velocity increases. 

(2) f decreases, with same velocity, as diameter increases. 

(3) f varies largely with the condition of the interior sur- 

face of the pipe, being larger the rougher the sur- 
face . 

In view of the last fact it will be seen that it is quite difficult 
to estimate a value for f which will hold for a particular pipe, 
as during a year ajid even from month to month the state of its sur- 
face will change. 

Merrimam in his Hydraulics gives a table showing prob- 
able values for f for clean cast-iron pipes based on discussions 
of Fanning, Smith and others. It is reproduced in Table VIII. 
An inspection of this table shows that a mean value is given by 
the decimal 0.02. Since f" varies with v, any problem in which 
V is unknown must be solved by first using a trial value for f , 
thus allowing v to be found approximately. With this value of v, 
a much closer value for f may be obtained to give a new v which 
will differ so little from the first as not to necessitate repeat- 
ing the operation. (Illustrative problems in class-room.) 

61. Critical Velocit y.- If a pipe contain water moving 
with a velocity higher than one which we will call the critical 
velocity, the frictional resistance will be found to vary nearly 
as the square of the velocity. For velocities below the critical, 
the resistance varies more nearly as the first power of the veloc- 
ity. Mesgrs. Saph and Schoder of Cornell University concluded, 
after maJcing a set of carefully conducted experiments, that the 
approximate value of the critical velocity for any particular pipe- 
might be represented by 

v_ = 0.1 



"- ^0.75 (56) 

d being the diam. in feet. From the equation it may be seen that 
for ordinary sizes of pipe the critical velocity has so low a 
value as to have no practical importance to the engineer. 

62. Loss by Bends .- If water that is flowing in a 
straight pipe be suddenly deflected from its course by a bend, 
it may be seen that by eddying and impact a loss of head may fol- 
low. Definite and complete information as to the size of loss 
and the manner of computing it is not available. Weisbach de- 
scribes some experiments made by him on small, smooth, iron pipes - 
bent in quarter circles. His resulteled him to propose 

Lost Head = c . v£ 

2g (57) 



-56- 



and he gives the following table for c, in which d is the diam. of 
the pipe and r the radius of the bend. 

For- d _ _go .40 .60 .80 1.0 1.2 1.4 1.6 1.8 2.0 

r 

c = .131 .138 .158 .206 .294 .440 .661 .977 1.40 1.98 

Weisbach failed to publish the sizes of his pipes but the above 
figures should probably be applied to very small pipes rather than 
large ones. 

In 1893-1898 Messrs. Williams, Hubbell and Fenkell car- 
ried on an elaborate series of experiments in the water mains of 
Detroit, Mich., and measured the losses of head in bends of 90", 
the pipes varying from 12 to 30 inches in diameter. The most re- 
markable thing in their conclusions was that, while Weisbach' s ex- 
periments indicated an increase in loss with increased sharpness 
of bends, the Detroit experiments showed a decrease in loss as the 
bends decreased in radius down to a limiting radius of 2-1/2 diam- 
eters. While at first glance these two sets of experiments seem 
to be contradictory it must be borne in mind that they were con- 
ducted under widely differing conditions. Weisbach' s bends being 
made in small pipes, the len gth of any curve itself was small and 
the loss occurring in it was principally that due to the deflec- 
tion of the water. William's pipes and curves were of such size 
that the length of the curve was large, running as high as 94 feet 
in one pipe. For such a curve the loss by pipe friction must have 
been a large part of the total loss, and the decrease in the loss 
of head by making the deflection on a gentler curve was more than 
offset by the increase in pipe friction. Using William's figures 
on a 30" pipe and calculating c for Weisbach' s formula (57) the 
following table has been derived. - 

For ^ = .042 .063 0.10 0.167 0.25 
r 

c = 1.36 .930 .738 .565 .390 

These figures should best apply to pipea from 1-1/2 to 3 ft. in 
diam. 

In view of the fact that our knov/ledge of the loss by 
bends is very insufficient, it is fortunate that in practical work 
the length of pipe is generally so long that the loss by a few 
bends is inappreciable compared with the greater loss by pipe 
friction. In a majority of cases the loss may be neglected. 

63.- Loss by Sudden Enlargement in Section .- If the cross- 
section of a pipe filled with flowing water be abruptly enlarged 
as in Pig. 62, the velocity will be reduced from v to v^ and a loss 



-57- 



of head will result from the impact and eddying caused by the meet- 
ing of the more swiftly moving water in the small pipe with the 

slower water in the large pipe. 
I I To properly estimate the loss 
we may use the principle of 
energy lost in direct inelastic 
impact, or proceed as follows. 
Let the area of cross-section 
^ in the pipes be a and a^ re- 
spectively, so that av = aiV]^. 
Writing Bernoulli's Theorem be- 
tween m and n we obtain 





f 


£ 


A 


-\ 1 1 




-m 




m 


n 


— j-zr 


_ 


>- 


p 


^r-o — 















D 


F 



r/^.62 



v^ p ^ 2 
2i + w + 0=Il + Pl + + 
2g ^ 



from which 



Lost Head = v^ 
2g 



Lost Head 
if - (£1 _ £) 

2g ^ w w^ 



(58) 



The velocity v is maintained up to and slightly beyond C-D so that 
at m' the pressure remains p. The total pressure in the large 
pipe on section C-D must then be a^^p while on E-F it is aip-i . 
That p-j^ is greater thaji p may be seen fi^om the fact that between 
m and n there occurs a large decrease in velocity-head without a 
corresponding gain in potential head. Consequently P]^/w must be 
greater than p/w. There exists therefore an unbalanced pressure 
between C-D and E-F against which W pounds of water move each sec- 
ond and thereby have their velocity changed from v to v^ . A 
measure of the resisting force being had from the relation. Force = 
Mass X Acceleration, we obtain 

aiPl - a^p = ^ (V - vij = ^^1^1 (v - vi) 
or 



^ - £ = II (V - vi) 



(59) 



Combining (58) and (59) 



2 

2g 2g 

Another fomi of expression may be derived from the relation 



Lost Head = v£ _ vi v^ (y - vi)' 

Set Per - -g ^^ - :^i; - — 2^ 



(60) 



-58- 



av = a^v-j^ . Substituting, in (60), for v its equal aiv^ 
there results 



Lost Head = ( — — " ^} 

~2g '^ 



= (^ - 1)M 



Sg (61) 



Both (60) and (61) are very important and should be remembered. 

The loss at entrance (Art. 58 and Fig. 59) may be close- 
ly determined by an application of equation (61) . Neglecting the 
loss at the sharp edge of the pipe and that due to pipe friction 
in the first 3 diame. (both being very small) it will be seen that 
the loss results from sudden expansion to fill the pipe. Assum- 
ing a contraction coefficient of 0.60 for the jet as it enters 
from the reservoir, equation (61) gives 

Lost Head = (1 " D^z! = o.44 v^ 
°*^ 2g ^ 

If to this a small amount be added to allow for losses neglected 
we may write 0.5 vS as was found in Art. 58. 
2g 

64. Loss by Sudden Contraction in Section .- If a pipe 
suffers a sudden contraction of section in the direction of the 

flow within, a loss of head results. 
Reference to Pig. 63 shows it to be 
divided into two parts. The ver- 
tical shoulder A-B causes impact and 
eddying in the end of the large pipe, 

ajid the water as it passes the sharp 

*"^ edge a-b contracts as from an orifice 
and subsequently suffers a loss of 
head in expanding to fill the pipe. 
The size of this loss will depend 
upon the ratio of the velocities 
and the amount of contraction at m, which is the same as saying 
it varies with the ratio of A^ to A (A]^ and A being respective 
areas of small and large pipe.) While a value may be approxi- 
mated by treating the loss as due wholly to the sudden expansion 
beyond m, a preferable method is to express the loss as equal to 
k.v2/2g and determine by experiment values of k for different 
ratios of Ai to A. Prof. L. M. Hoskins in his recent (1906) 
work on Hydraulics gives the following convenient table which is 
based on data from Weisbach. 




/7^.63. 



-59- 



For Ai = ,10 .20 .30 .40 .50 .60 .70 .80 .90 1.00 
A 

k = .362 .338 .308 .267 .221 .164 .105 .053 .015 

65. Summary of Losses in Pipes. - For ready reference the 
different losses and their values are here summarized. - 
(1) Head Lost at Entrance = 0.5v2 



= 


f 1 v2 
d 2g 








= 


c.yS 
2g 








= 


{v-vi)2^ 


:(aj. 


■1) 


2 
VI 

Sg 


= 


k.v2 
2S 









2g 
(8) Head Lost by Pipe Friction 

(3) Head Lost by Bends 

(4) Head Lost by Sudden Enlargement 

(5) Head Lost by Sudden Contraction 

If a pipe be in einy way obstructed, the resulting lose may be ap- 
proximately determined from (4) by letting a^ represent the area 
of the pipe and a the area at the point of obstruction. 

The general equation (50) may be now thus elaborated. 

h = v2 + .5v2 + f 1 x£ + Losses by bends, etc. (62) 
2g '~2g' ^ 2g 

.and if ' the pipe be fairly straight and of uniform section through- 
out 

h = v2 + .5v2 + f 1 v£ 

2i ~^i~ d 2g (63) 

In determining the velocity in a ver y lon g pipe it will be found 
that the loss by friction is so nearly the entire head h that we 
may write 

_ j£ 

d 2g . (64) 



h = -f i ^^ 



without an error greater than that liable to result in the selec- 
tion of a value for f . 



-60- 



66. Solution of Pipe Problems .- In the above equations, 
the principal variables are h, 1, d and v. A majority of the 
pipe problems arising in practice consist in having one of these 
linicnown and desired. The solution may be effected by using 
either (62), (63) or (64), as the case requires, and solving di- 
rectly for the unknown. Two very common problems are 



(1) Given h, 1 ajid d to find v and Q. 
representative, we have 



Choosing (63) as 



V = 



N 



.2^ 



1.5 + f 1 



and Q = Av 



(65) 



where A is the area of the pipe, = trd£ 

4 * 

(2) Given h, 1 and ^ to find d necessary to deliver Q. Again 
using (63) and remembering that v = 4Q we have 

iTd^ 



d = 



1.5 + fl 



2gh 



(f)' 



(66) 



In Prob . 1 a tentative value of 0.02 is first assumed for f 
and then corrected to accord with the resulting value of v. 
The process is repeated until a value for v is obtained which 
requires no new value for f . 

In Prob. 2, f is taken again as 0.02 and from the resulting d 
a value of v is obtained by which to select a closer value of 
f . 



Fi0.64r 



67. Piezometers. - To measure the pressure of water 
against a pipe wall it is customary to use an open tube which is 

inserted through the wall, 
normal to its inner surface, 
and extended vertically upward 
on the outside. (Fig. 64). 
Water rises in the tube and, 
as its level Will be at rest, 
the .height of the coliimn will 
be an exact measure of the 
pressure at its base. Such a 
gage is called a piezometer. 
Certain precautions must be 
taken in using it to avoid 
inaccuracies. Pig. 65 shows 




-61 



three tubes, of which (a) is normal to the inner surface. 

It measures the pres- 
sure at its base cor- 
rectly. Both (b) and 
(c) are inclined to the 
inner surface although 
their edges are flush 
with it. The water in 

(b) will stand lower 
than in (a) while in 

(c) it will stand high- 
er. Both give incor- 
rect readings on account 
of their columns being 
affected by the velocity 

of the moving water. Had their lower edges not been flush with 
the inner surface, the discrepancies might have been further mag- 
nified. 




F/g.6S. 



68. Piezometer Measurements. - Hydraulic Gradient. - Fig. 
66 shows a piezometer inserted at A in a straight pipe of uniform 
bore which discharges into the air. Between m and n we may write 




F/0. 66. 



+ Pg^ + h=v£+ (£ + Pa 
If 2g ''w w 



) + + Lost Head 



p/w being the height of the water colunin above n, and h the dis- 
tance of n below the reservoir level. Evidently p is the "rela- 
tive pressure" and would be zero if n were at the 'outlet of the 
pipe. (See Pig. 58 and equation (50). More simply may the above 
be written 



h = v£ + £ + Lost Head 
2g w 



(67) 



euid it should be noted that h is divided into three parts. 



-62- 



Since one part, p/w, is the height of the water column, the sura of 
the other two is the distance down from the reservoir level to the 
top of the column. At any point below n, such as n' , the velocity 
head will be unchanged but the loss in head will be increased by 
reason of the loss by pipe friction in the intervening distance. 
Hence the svm of these two heads will be larger than at n ajid the 
level of the second water coliJimn stands lower than the first. 
The difference in level being hi we may write 



hi = f ^ 



1 v^ 



d-2g 



(68) 



from which, if h]^ and 1 be carefully measured, the value of v may 
be found. A tentative value of 0.02 must be used in first solv- 
ing for V and may be then corrected and recorrected as explained 
in Art. 60. 

The equation also shows that in a straight pipe (or ap- " 
prox. straight) the loss in head between two points, and conse- 
quently the drop in piezometer levels, is proportional to the 

length of pipe inter- 

-j 11 ,| 1 vening. Therefore if 

"" a row of piezometers 

were inserted along the 
pipe from reservoir to 
outlet, a straight line 
would join the levels 
of their water columns. 
(Fig. 67) It would 
start (practically) from 
a point B, a little be- 
low the reservoir level, 
and run to C at the outlet. The distance A-B represents v2/2g + 
Lost Head at Entrance, and the line must run to C since the pros- 
sure there is only atmospheric. This line is commonly known as 
the Hydraulic Grade Line or Gradient and 'a vertical ordinate be- 
tween any point in the pipe line and the Gradient will measure the 
pressure head at that point. If the pipe be quite long, the 
slope of the Gradient may be approximately written 




/%.d7 



1 



(69) 



and slight changes in vertical elevation may be made by gentle 
curves without sensibly changing the Gradient from a straight line. 
(This follows from the fact that the length 1^ would still remain 
practically the same as its horizontal projection.) 

If a pipe be laid so that a portion of it comes above 
the Gradient a serious disturbance may result in the flow. At 
any point n, Fig. 68, in the part lying above the Gradient, the 



-63- 



preasure must be less thaai atmospheric since the vertical ordinate 
must be measured downward to the Gradient. If the pipe be abso- 




fy^. 66 



lutely air tight, flow will take place as in the ordinary case. 
If air can gain entrance to the pipe by Joints or other means, 
then it will collect at the summit near n and the pressure will 
approach the atmospheric . If this condition of pressure be ob- 
tained the gradient will shift to A-C and the discharge will be 
due to the head h' . The portion of the pipe between C and B will 
act as a channel to carry off the decreased flow from C. 

The importance of knowing the relative positions of the 
Gradient and pipe line is at once seen. No long pipe line having 
a small total fall and sensible variations in level should be laid 
in the ground until a Gradient is plotted on the profile of its 
course and the absence of "summits" above the Gradient assured. 
In pipes where summits do exist, flow may be maintained by placing 
pumps at these points which shall exhaust the collected air at in- 
tervals during the period that flow must be maintained. 



General Expression for Loss in Head .- Much has been 
chapter concerning the different losses of head occur- 
pe line during flow. It will be of interest before 
subject to derive a general expression, covering the 

experienced by water passing through any discharging 
erms of the velocity coefficient of the device and the 
discharge. To make this clear let us assume a noz- 



69 
said in this 
ring in a pi 
leaving the 
losB in head 
device, in t 
velocity of 

zle, having a coefficient of velocity, Cy, to discharge under a 
head h. 



Theoretically 



= -^[2ih 



and 



v^ = h 
2g 



and 



Actually 
V = c^Sgh 

v£ = c|h 

2g 



The value of the lost head must be 



Lost Head = h - c^h = (1 - 



c^)h 



If for h its value in terms of actual v be substituted there re- 
sults 



-64- 



Lost Head = ( 1 - Cy ) v£ = /_1 _ ]^\ v; 



2g 



2g 



(70) 



We have assumed a nozzle for purpose of discussion but it is evi- 
dent that this important equation may be applied to any device. 

70. Pipe Line and Nozzle .- A nozzle is often attached to 
a pipe line where the water is to be delivered to a power wheel or 
used for industrial purposes. Fig. 59. illustrates a general con- 
dition. The general equation (63) enables us to write 




h = v£ + 
2g 



.5 



2g 



Fi^. 69. 

+ f l.vf 
^ 2i 



- 1 



) v< 



We have two tmknowns, v and v-|^, but also the relation 

_v = d£ 



so that the equation may be solved. 

The Gradient is shown for pipe and nozzle. It will be 
seen that the pressure at the base of the nozzle is considerable, 
due to the much slower velocity at that point. If the energy of 
the issuing jet is to be utilized for power, it is desirable that 
the losses in head between reservoir and outlet be reduced to a 
mlnimiim. Evidently abrupt chajiges in pipe sections are to be 
avoided and, to reduce pipe friction, the diam. of the supply pipe 
should be large . 

71. Branching Pipes .- Fig. 7o. shows a problem arising 
in water-works design. A high-level reservoir supplies two oth- 
ers at lower levels by means of a main and branches. Let it be 
desired to find the rate of discharge into each reservoir, the 
length and diameter of each pipe being known, also the levels of 
the reservoir. 

Let it first be observed that the water, in passing 
t hrough the main , experiences a loss in head due to pipe friction, 
(other losses neglected) . Each particle of water, on leaving the 



-65- 



junction, has a certain total head (v"/8g + p/w + z) a part of 
which must be lost in friction as it goes through the branch to 




the lower reservoir. Therefore for a particle starting at m and 
arriving at ni we may write 



Hi = 



f 1 v^ 



f ll V 



(see equation 50) 



1 
2i " d 2g d^; Sg 

and for a particle leaving m ajid arriving at n2 we have 

,2 



Hs = v2 



_2 + 

2g 



f 1 XI 
d 2g 



f i2 ;^ 

^2 2s 



In these two equations appear the three unknowns, v, v-i and Vr, . 
Evidently another equation is necessary for their determination. 
This is furnished by the "equation of continuity" 



av = a-j^v-j^ + a 



^2^2 



when a, a^ and ao are the areas of the pipe sections and are known. 
As in previous pipe problems, it is necessary to assume f = 0.02 
in the first solution for v, v-^ and Vg and correct according to 
the values resulting. 



-66- 



CHAPT5R V 

Flow in Open Channels . 

72. Divialona of Subject .- It is not the intention here' 
to treat of all the many and varied problems presenting themselves 
under this subject. Broadly viewed, all problems of flow in open 
channels may be classified under tne following di visions. - 

1. Plow in Natural Channels 

S. Flow in Artificial Channels 

(a) Uniform flow 

(b) Non-uniform flow. 

75. Natural Channels .- The discussion of natural channel 
flow will not be given here. While much has been done in this 
important field of hydraulics, all efforts to originate an accurate , 
or reasonably accurate, formula for flow which will apply to 
streams of all kinds and sizes, so far have come to naught. The 
conditions are too complex and varied to hope that success may be 
yet attained. 

The velocity and discharge of natural streams may be best 
obtained by actual measurements made in the field. For this pur- 
pose various forms of current meters and floats have been devised 
and quite successfully used. A description of these and the 
methods followed in making a gaging are well set forth in a re- 
cent(l907) work on "River Discharge" by Messrs. J. 0. Hoyt and 
N. C. Grover of the U. S. Hydrographic Bureau. 

74. Artificial Channels.- Uniform Flow .- This is the case 
with which the engineer has mostly to do. The assumption of 
"steady flow" is made, so that the quantity of water Q, passing a 
transverse section of the stream is constant. To make the flow 
" uniform ." the velocity of flow past such a section must be con- 
stant for all positions of the section. This necessitates that 
the cross-sectional area of the stream be constant, so that along 
any longitudinal section we must have a constant depth. The sur- 
face is therefore parallel to the bed, both having some angle of 
inclination o< with the horizontal. We call this inclination the 
"slope" of the channel and express it as 

s = h 
1 

h being the vertical fall in the length of channel 1, and s the 
sine of Of. (Pig. 71) 

In any cross-section A,B,C,D that part of the outline of 
the section which comes in contact with the bed and bajiks (A-B-C-D) 
is known as the "wetted perimeter" and will be designated as p. 
In the discussion which follov/s there occurs the quantity A/p7 A 



-67- 



being the area of the streajn's cross-section. It will be conve- 
nient to refer to it by name, and that of "Hydraulic Radius" or 



7777. 




n^.7/. 




"Hydraulic Mean Depth" ia generally given it. We will represent 
it by r. 

With the above assumptions-, as to flow, made, we may re- 
gard the water between any two sections, A-B and E-P (Fig. 71) 
as a solid prism of water having a uniform motion down the in- 
clined trough of the stream. The forces producing or hindering 
motion are its weight. Awl, the end pressures, V-^ and Pg, and the 
frictional resistance, R, offered by the sides of the trough. 
(Pig. 72). The forces P]^ and P^ must be equal and opposite as 

they represent the pressures on 
two equal areas .under the same 
7 , head. We need not consider them. 

' ■ The component of the weight W 
along the line of motion being 
Awl.sineo<, we have 

Awl .sine o< - R = 




i\lV=^iv/ 



F = Awl .sineo < 
Pl 



If F be the value of the fric- 
tional resistance per unit of 
area, so that R = P x pl, the 
above may be written 

or F = w.r.s 



(71) 



Our knowledge of fluid friction as outlined in the previous chap- 
ter (Art. 59) shows F to equal approximatel y some function c^ of 
the velocity squared. (See Critical Velocity, Art. 61) If then, 
we replace P by cv2 and solve for v we obtain 



V = C\|rs 



(72) 



in which large C has replaced ^w/c . This formula is known as 
Chezy's Formula and is one of many that have been proposed. Its 



-68- 



derivation up to and including equation (71) w^b rational and 
sound in theory. There rational treatment ceased and the assump- 
tion that P = cv^ was really only a rough truth. If it were 
strictly true , C should be constant for any particular channel 
varying only with the roughness of its lining. Experiments would 
indicate, however, that it is not so, but varies also with the 
slope and hydraulic radius. Entire dependence must therefore be 
placed on the values of C as determined by experiment. 

In Art. 68 it was shown that the velocity of flow in a 
pipe might be figured from 

where hi is the difference in level of two piezometer columns in- 
serted in a pipe a distance apart equal to 1. That this is only 
another form of Chezy's Formula may be seen if we express d in 
terms of r, the hydraulic radius, and equate for v. For a circu- 
lar pipe filled with water r would equal 1Td2/4 -^iTd or d/4, from 
which d = 4r. Making the substitution and transposing we obtain 



V=J8£ 

or 



{-1^ N" I 



r X h 



v = C -^ rs 

In this formula b is the slope of the Hydraulic Gradient, while in 
equation (72) ^ is the slope of the water surface in the open chan- 
nel. It is quite evident that for an open channel the slope of 
the Hydraulic Gradient and the water surface are identical as the 
free water surface would mark the levels of all piezometer columns 
placed in the channel . The two formulae are therefore identical . 

75. Determination of the Constant C - An inspection of 
Chezy's formula shows that C is not an abstract number, but de-- 
pends upon a distance. It will vary in value then for different 
systems of units. From experiment we know it also varies with 
the roughness of the channel lining, with r and with e. The 
chief determining factor is the roughness of lining. As flexi- 
ble limits to its value under different conditions of the latter 
the following are given. 

For timber lined conduits 

For smooth masonry conduits 

For rough masonry conduits 

For ditches in clean earth or gravel 

For ditches filled with weeds or rubbish C = 

Values for C may be determined experimentally by measuring v, r 



C = 100 


to 


125 


C = 85 


to 


110 


C = 60 


to 


90 


= 50 


to 


"80 


C = 20 


to 


40 



-69- 



and 8 and computing C from the formula. Of course, v and b may- 
be difficult to determine accurately, the latter being apt to vary 
from point to point in any canal. 

76. Bazin's Formula for C- Bazin made a careful study 
of many experiments and proposed the formula, - 



C = 



87 



0.552 + m 

when m has the following values, - 

For smooth cement or boards m = 0.06 

For timber or brick lining m = 0.16 

For masonry lining m = 0.46 

For ditches in clean earth m = 1.30 

For ditches in bad condition m = 1.75 

This formula gives results that are fairly satisfactory 
and is in common use in countries abroad. 

77. Formula of Ganguillet and Kutter .- In 1869 Messrs. 
Ganguillet ajid Kutter made a most comprehensive study of all the 
reliable experiments which had been made up to that time and from 
them deduced a formula for C which is the only general rule of 
wide application that we have. The experiments from which they 
drew their conclusions ranged from observations on small artificial 
channels up to measurements made on the Mississippi River. Their 
formula is as follows,- 

1^811 + 41.65 + 0-QO^Q^ 
C = (73) 



I pi. 65 + O- 00281 ] 



This makes C a function of thrpe quantities,- £ the slope, 
r the hydraulic radius and n, an abstract number known as the co - 
efficient of roughness . Values for the latter are given by Gan- 
guillet and Kutter as follows,- 

- Values of n in Kutter' s Formula - 
n = .009 for well planed timber 
.010 for neat cement 
.011 for cement with i/3 sand 
.012 for unplajied timber 
.013 for ashlar axid brick 



-70- 



.015 for canvas lining on frames 

.017 for rubble masonry 

.020 for canals in very firm gravel 

.025 for canals and rivers in perfect order 

.030 for canals and rivers in fair order 

.035 for canals and rivers in bad order and strevm 

with detritus or overgrown with vegetation 

While Kutter's Formula gives results that are wonder- 
fully good considering its generality, the student should not 
expect a high degree of accuracy in its use, even under most fa- 
vorable conditions. An error as large as 10 per cent, may be 
regarded as liable to occur from its use. 

IVhere constant use is made of this formula, time and 
labor may be saved by the aid of a set of diagrams based upon it, 
prepared and published by Prof. I. P. Church or Cornell Univ. 
From these, values of v may be directly taken for assigned values 
of r, B_ 6Lnd n. 

78. Most Advantageous Cross Section. - If- an open channel 
has its slope £ and wetted croos-sectional area A fixed , it is evi- 
dent that the majcimum velocity, (hence meix. discharge) will occur 
when the area A is so proportioned that the surface of contact be- 
tween water and channel lining will be as small as possible. 
(Frictional resistance being reduced to a minimum.) This is the 
same as saying that p must be a minimum. It may be further seen 
that this condition will also permit using the least p ossible slo pe 
consistent with the demand that a fixed Q, be discharged. Therefore 
in general a very efficient section is obtained by making p a min- 
imum. Remembering that r = A/p, the making of r a maximijm obtains 
the same result . 

Since of all figures having equal areas the circle has the 
least perimeter, it follows that a semi-circular section has the 
smallest possible value of p. (Fig. 73(a). Such a section. is 
difficult to build and mainTain under some conditions and general- 
ly the trapezoidal or rectcingular form is used.' Since of all 



t 




/w^.7J 




-J—)^ 



(CJ 



1 



rectangles the square has the least perimeter for a fixed area, 
it will be advantageous to make the rectangular section a half 
square. (Fig. 73c). Similarly if it is to be trapezoidal, p will 



-71- 



be least for a half -hexagon. (Pig. 73b). 

For any other form of section, the best proportions may 
be fo\and by expressing r in terms of the fixed area and one of its 
dimensions. The putting of the first differential of this ex- 
pression equal to zero will permit of the solution of the equation 
for the value of this dimension. This may be illustrated as fol- 
lows.- 

We will suppose it is desired to find the relation be- 
tween d and b (for "best section") for the channel shown in Pig. 
74, maintaining a fixed area of 100 sq. ft. We have 




(a) 
(b) 

(c) 



A = (b + 2d)d = 100 



r = 



100 



b + 2dvJ^ 

b = 100 - 2dfi 
d 



Combining (c) and (b) we have 

J. = lOOd 

100 - 2d2(l - xl2) 

and placing dr/dd = there results d = 5.95 ft . 
From (c) we then obtain b = 4.92 ft . 

79. Non-uniform Flow. - If in axiy channel the slope and 
cross-section be variable, it will be found that, although the flow 
is "steady," giving for all sections the relation 

Q, = a^vi = agvg = ajVg = etc . 

the mean velocit y of the stream varies from section to section. 
The case is one of " non-uniform " flovi. While it offers certain 
practical problems which have their importajice to the hydraulic 
engineer, its relative importance is not such as to warrant a dis- 
cussion of it in this limited course. (For references, see Mer- 
rimam's "Hydraulics" or Church's "Mechanics of Engineering.") 



-72- 

CHAPTER VI 

Force and Energy in Jeta 

80. Force Acting upon a Jet .- In the case of a jet issu- 
ing from an orifice, the velocity v , which all the particles are 
assumed to have, must be due to the action of a force or pressure 
exerted by the water in the reservoir upon the particles at the 
base of the jet. The magnitude of the force is such that in each 
small interval of time At, a small mass of water ZlM has its veloc- 
ity changed from to v. Its mean acceleration is therefore 

a = V - Q 

so that the measure of the force becomes 

P = ^M X _v 
Zlt 

If W be the weight of water discharged in a unit time, thenz]M = 
W.Zlt and 

P = W./dt X _v = Wv 

g -^t g (74) 

Since W = w.a.v, another form of expression would be 

P = wav^ = 8awh (75) 

S 

From this it appears that the force is twice the static pressure 
that would exist on a plug just filling the orifice. This con- 
dition is a theoretic one, however, since in deriving (75) from 
(74) no allowance was made for the contraction of the jet or for 
the effect of friction upon the velocity. For a standard orifice, 
assuming Cy = .99 > °c ~ '^^ ^^^ °d ~ '^^' ^® have 



W = .62wav and v = . 98\j2gh 

l£ = (.98)^h 
2g 

v being the actual velocity in the contracted section. Under 
these conditions 

P = . 6gwav^ = 2 LeSwa x (.98)^hn = 1.19awh 
S . 
Mr. Peter Ewart of England made an actual measurement of P using a 
standard orifice and obtained 



-73- 
P = 1 . 14awh 
which is a satisfactory verification. 

81. Reaction of a Jet.- If the water in the reservoir 



Fig 7-5 



exerts a pressure P upon the base of the jet, it must in return 

receive a like pressure from the jet 
which it transmits to the walls of the 
reservoir. If we mount the reser- 
voir on wheels (Pig. 75) this force P 
will cause it to move to the left with 
an accelerated motion, provided the 
frictional roaistance be smaller than 
the force. P in this instance is the 
" Reaction " of the jet. 

82. Energy in a Jet .- If a jet moves with a velocity v 
and discharges W pounds of water per second, it possesses an amount 
of Kinetic Energy which may be expressed by 




7777 



K = W y£ 
2g 



(77) 



That this is so may be seen if we consider that the velocity v is 
due to a free fall through the height h.( = v^/sg) . The poten- 
tial energy Wh, which W pounds of water have at the initial state 
of rest is changed without loss into Kinetic Energy. Equation 
(77) is in the usual form 

K = Mv£ 
2 



85. Force Exerted by a Jet upon a Deflecting Surface. - 
If a jet be turned from its path by meeting tangentially a deflect- 
ing surface, (Fig. 76) it exerts upon the surface a dynamic pres- 
sure P. We may consider the equal 
and opposite force P' to have been 
the cause of the deflection and 
measure it as follows. Assuming 
the surface to be smooth, the -jet 
will pass over it with undiminished 
velocity. In each small interval 
of time Zlt a mass ^3M has its veloc- 
ity in the initial direction of the 
jet changed from v to v coso(. Its 
mean acceleration in this direction 
is therefore 




V7Z777777V77777// 



o - V - V coso< 
^ Zt ~ 



-74- 



The measure of the accelerating force (acting parallel to X axis) 
ia found from 

H = ' ^^ (v - V coacx ) 

and since ZIM = W ^idt we may write 

S 

H ^ W.v(l - ooscx) ^^Q) 

~ g 



Similarly we may prove 



y _ W. V sineo< 



(79) 



For value of P' we have 



/v^.7Z 



H 



H 



P' = ^v2 + H^ = W ^ J2(l - cosc'<) 



(80) 






84 



. Special Cases, (l) Flat Plate Perpen 
dicular to Jet.- Here «< = 90 a 



and 



P is the X component of pressure, 
From (78) 

p = W.v 
g 



^>. 



Case (g) Jet turned through 180 - (Fig. 73) If the 
surface be formed so as to cause the jet to be turned completely 
back upon itself we havec<= 180° and 
P ia the sum of the X components of 
pressure . 

• p = W.v [I - i-lj] = 2 W V 
" g g 

These two cases are of interest as fur- 
nishing a basis for discussing the force 
exerted by a jet upon the moving vanes 
of a water wheel. 

Problem.- Let the student find the value of P in each of 
the above cases, assuming the vanes to have a velocity vq parallel 
to, and away from, the jet. 

For a further study of moving vanes see Church's "Hydrau- 
lic Motors." 




-75- 





Table I 


Weight of Distilled 


Water. 




Temperature 


Pounds per 


Temperature 


Pounds per 


Temperature 


Pounds per 


Fahrenheit. 


Cubic li'oot 


Fahrenheit. 


Cubic Foot 


Fahrenheit . 


Cubic Foot 


32 


62.42 


95 


62.06 


160 


61.01 


35 


62.42 


100 


62.00 


165 


60.90 


39.3 


62.424 


105 


61.93 


170 


60.80 


45 


62.42 


110 


61.86 


175 


60.69 


50 


62.41 


115 


61.79 


180 


60.59 


55 


62.39 


120 


61.72 


185 


60.48 


60 


62.37 


125 


61.64 


190 


60.36 


65 


62.34 


130 


61.55 


195 


60.25 


70 


62.30 


135 


61.47 


200 


60.14 


75 


62.26 


140 


61.39 


205 


60.02 


80 


62.22 


145 


61.30 


210 


59.89 


85 


62.17 


150 


61.20 


212 


59.84 


90 


62.12 


155 


61.11 









Table II 


. Coeff 


icientB . 


For Circular Orifices 


. 


Head 
h 
in Feet. 


Diameter of Orifice in Feet 


0.02 


0.04 


0.07 


0.1 


0.2 


0.6 


1.0 


0.4 




0.637 


0.624 


0.618 








0.6 


0.655 


.630 


.618 


.613 


0.601 


0.593 




0.8 


.648 


.626 


.615 


.610 


.601 


.594 


0.590 


1.0 


.644 


.623 


.612 


.608 


.600 


.595 


.591 


1.5 


.637 


.618 


.608 


.605 


.600 


.596 


.593 


2.0 


.632 


.614 


.607 


.604 


.599 


.597 


.595 


2.5 


.629 


.612 


.605 


.603 


.599 


.598 


.596 


3.0 


.627 


.611 


.604 


.603 


.599 


.598 


.597 


4.0 


.623 


.609 


.603 


.602 


.599 


.597 


.596 


6.0 


.618 


.607 


.602 


.600 


.598 


.597 


.596 


8.0 


.614 


.605 


.601 


.600 


.598 


.596 


.596 


10.0 


.611 


.603 


.599 


.598 


.597 


.596 


.595 


20.0 


.601 


.599 


.597 


.596 


.596 


.596 


.594 


50.0 


.596 


. .595 


.594 


.594 


.594 


.594 


.593 


100.0 


.593 


.592 


.592 


.592 


.592 


.592 


.592 



76. 



Table III Coefficients for Square Orifices. 



Head 
h 
in Feet. 




Side of the Square in 


Feet. 






0.02 


0.04 


0.07 


0.1 


0.2 


0.6 


1.0 


0.4 




0.643 


0.628 


0.621 








0.6 
0.8 


0.660 
.652 


.636 
.631 


.623 
.620 


.617 

.615 


0.605 


0.598 
.600 


0.597 


.605 


1.0 


.648 


.628 


.618 


.613 


.605 


.601 


.599 


1.5 


.641 


.622 


.614 


.610 


.605 


.602 


.601 


2. 


.637 


.619 


.612 


.608 


.605 


.604 


.602 


2.5 


.634 


.617 


.610 


.607 


.605 


.604 


.602 


3. 
4. 


.632 
.628 


.616 
.614 


.609 
.608 


.607 
.606 


.605 
.605 


.604 
.603 


.603 


.602 


6. 


.623 


.612 


.607 


.605 


.604 


.603 


.602 


8. 


.619 


.610 


.606 


.605 


.604 


.603 


.602 


10. 


.616 


.608 


.605 


.604 


.603 


.602 


.601 


20. 


.606 


.604 


.602 


.602 


.602 


.601 


.600 


50. 


.602 


.601 


.601 


.600 


.600 


.599 


.599 


100. 


.599 


.598 


.598 


.598 


.598 


.598 


.598 



Table IV Coefficients for Rectangular Orifices 1 Foot Wide, 



Head 
h 
in Feet. 




Depth 


of Orifice in Feet 






0.125 


0.25 


0.50 


0.75 


1.0 


1.5 


2.0 


0.4 


0.634 


0.633 


0.622 










0.6 


.633 


.633 


.619 


0.614 








0.8 


.633 


.633 


.618 


.612 


0.608 






1. 


.632 


.632 


.618 


.612 


.606 


0.626 




1.5 


.630 


.631 


.618 


.611 


.605 


.626 


0.628 


2. 


.629 


.630 


.617 


.611 


.605 


.624 


.630 


2.5 


.628 


.628 


.616 


.611 


.605 


.616 


.627 


3. 


.627 


.627 


.615 


.610 


.605 


.614 


.619 


4. 


.624 


.624 


.614 


.609 


.605 


.612 


.616 


6. 


.615 


.615 


.609 


.604 


.602 


.606 


.610 


8. 


.609 


.607 


.603 


.602 


.601 


.602 


.604 


10. 


.606 


.603 


.601 


.601 


.601 


.601 


.602 


20. 








.601 


.601 


.601 


.602 



77. 
Table V Coefficients for Submerged Orifices. 



Effective 
Head in Feet. 


The Size 


of Orific 


e in Feet 






Circle 


Square 


Circle 


Square 


Rectangle 




0.05 


0.05 


0.1 


0.1 


0.05x0.3 


0.5 


0.615 


0.619 


0.603 


0.608 


0,623 


1.0 


.610 


.614 


.602 


.606 


.622 


1,.5 


.607 


.612 


.600 


.605 . 


.621 


2.0 


.605 


.610 


.599 


.604 


.620 


2.5 


.603 


.608 


.598 


.604 


.619 


3.0 


.602 


.607 


.598 


.604 


.618 


4.0 


.601 


.606 


.598 


.604 





Table VI Coefficients for Contracted Weirs, 



Effective 




Length of Weir 


in Fef 


5t 


• 


Head 


























in Feet. 


0.66 


1 


2 


3 


5 


10 


19 


0.1 


0.632 


0.639 


0.646 


0.652 


0.653 


0.655 


0.656 


0.15 


.619 


.625 


.634 


.638 


.640 


.641 


.642 


0.2 


.611 


.618 


.626 


.630 


.631 


.633 


.634 


0.25 


.605 


.612 


.621 


.624 


.626 


.628 


.629 


0.3 


.601 


.608 


.616 


.619 


.621 


.624 


.625 


0.4 


.595 


.601 


.609 


.613 


.615 


.618 


.620 


0.5 


.590 


.596 


.605 


,608 


.611 


.615 


.617 


0.6 


.587 


.593 


.601 


.605 


.608 


.613 


.615 


0.7 




.590 


.598 


.603 


.606 


.612 


.614 


0.8 






.595 


.600 


.604 


.611 


.613 


0.9 






.592 


.598 


.603 


.609 


.612 


1.0 






.590 


.595 


.601 


.608 


.611 


1.2 






.585 


.591 


.597 


.605 


.610 


1.4 






.580 


.587 


.594 


.602 


.609 


1.6 








.582 


.591 


.600 


.607 



78. 





Table VII Coefficients 


for Suppressed Weirs 


• 


Effective 






Length of Weir : 


Ln Feet 






Head 
in 




























Feet 


19 


10 


7 


5 


4 


3 


2 


0.1 


0.657 


0.658 


0.658 


0.659 








,0.15 


.643 


.644 


.645 


.645 


0.647 


0.649 


0.652 


0.2 


.635 


.637 


.637 


.638 


.641 


.642 


.645 


0.25 


.630 


.632 


.633 


.634 


.636 


.638 


.641 


0.3 


.626 


.628 


.629 


.631 


.633 


.636 


.639 


0.4 


.621 


.623 


.625 


.628 


.630 


.633 


.636 


0.5 


.619 


.621 


.624 


.627 


.630 


.633 


.637 


0.6 


.618 


.620 


.623 


.627 


.630 


.634 


.638' 


0.7 


.618 


.620 


.624 


.628 


.631 


.635 


.640 


0.8 


.618 


.621 


.625 


.629 


.633 


.637 


.643 


0.9 


,619 


.622 


.627 


.631 


.635 


.639 


.645 


1.0 


.619 


.624 


.628 


.633 


.637 


.641 


.648 


1.2 


.620 


.626 


.632 


.636 


.641 


.646 


. 


1.4 


.622 


.629 


.634 


.640 


.644 


, 




1.6 


.623 


.631 


.637 


.642 


.647 







Table VIII Values of Friction Factor for Clean Iron Pipes, 



Diameter 

in 

Feet 






Velocity 


in Feet 


per Second 




1 


2 


3 


4 


6 


10 


15 


0.05 


0.047 


0.041 


0.037 


0.034 


0.031 


0.029 


0.028 


0.1 


.038 


.032 


.030 


.028 


.026 


.024 


.023 


0.25 


.032 


.028 


.026 


.025 


.024 


.022 


.021 


0.5 


.028 


.026 


.025 


.023 


.022 


.020 


.019 


0.75 


.026 


.025 


.024 


.022 


.021 


.019 


.018 


1. 


.025 


.024 


.023 


.022 


.020 


.018 


.017 


1.25 


.024 


.023 


.022 


.021 


.019 


.017 


.016 


1.5 


.023 


.022 


.021 


.020 


.018 


.016 


.015 


1.75 


.022 


.021 


.020 


.018 


.017 


.015 


.014 


2. 


.021 


.020 


.019 


.017 


.016 


.014 


.013 


2.5 


.020 


.019 


.018 


.016 


.015 


.013 


.012 


3. 


.019 


.018 


.016 


.015 


.014 


.013 


.012 


3.5 


.018 


.017 


.016 


.014 


.013 


.012 




4. 


.017 


.016 


.015 


.013 


.012 


.011 




5. 


.016 


.015 


.014 


.013 


.012 






6. 


.015 


.014 


.013 


.012 


.011 







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